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Linear Algebra: Matrices - Assignment | Mathematical Methods - Physics PDF Download

Q.1. If θ - ∅ = an odd multiple of — find the product of matrices of Linear Algebra: Matrices - Assignment | Mathematical Methods - Physicsfind the product of matrices
Linear Algebra: Matrices - Assignment | Mathematical Methods - Physics
Ans.The product of A and B is
Linear Algebra: Matrices - Assignment | Mathematical Methods - Physics
Linear Algebra: Matrices - Assignment | Mathematical Methods - Physics
Linear Algebra: Matrices - Assignment | Mathematical Methods - Physics
Since d and S differ by an odd multiple of Linear Algebra: Matrices - Assignment | Mathematical Methods - Physicstherefore θ - ∅ = an odd multiple of Linear Algebra: Matrices - Assignment | Mathematical Methods - Physics
thus, cos (0-0) = 0. Hence, Linear Algebra: Matrices - Assignment | Mathematical Methods - Physics

Q.2. Find the values of a, b, c, d, e, f where a, b, c and f are positive and d and e are negative such that the matrix Linear Algebra: Matrices - Assignment | Mathematical Methods - Physicsis orthogonal.
Ans. In an orthogonal matrix the sum of the squares of elements in any row or any columns is unity. Here, Linear Algebra: Matrices - Assignment | Mathematical Methods - Physics
Linear Algebra: Matrices - Assignment | Mathematical Methods - Physics
Linear Algebra: Matrices - Assignment | Mathematical Methods - Physics
⇒ b2 + c2 = 5 and b+2c = 4
solving these two relation we obtain b = 2 and c = 1
Linear Algebra: Matrices - Assignment | Mathematical Methods - Physics
Linear Algebra: Matrices - Assignment | Mathematical Methods - Physics

Q.3. If A is a matrix such that
Linear Algebra: Matrices - Assignment | Mathematical Methods - Physics
Find the matrix A20 + A15 + 2I, where I is 3 x 3 Identity matrix.
Ans. Linear Algebra: Matrices - Assignment | Mathematical Methods - Physics

Thus, A2 = I, A3 = A, A4 = I, A5 = A and so on.

Thus we see that the odd power of A are equal to A and even powers of A are equal to I. Hence    A20 + A15 + 21 = I + A + 2I = A + 3I
Linear Algebra: Matrices - Assignment | Mathematical Methods - Physics

Q.4. The trace of a 3 x 3 matrix is 12 and the determinant of the matrix is 48 . The product of the smallest and the largest eigenvalues is 12. If all the eigenvalues of the matrix are different then

(a) Find the eigenvalues of the matrix.

(b) Write the chatacteristics equation.
Ans. λ1 + λ2 + λ3 = 12, λ1λ2λ3 = 48
Let A be the smallest eigenvalue and λ3 be the largest eigenvalue .
Then, λ1λ3 = 12 ⇒ λ= Linear Algebra: Matrices - Assignment | Mathematical Methods - Physics
Hence, λ+ λ3 = 8
1 - λ3)2 = (λ1 + λ3)2 - 4λ1λ3  = 82 -4.12 =16 ⇒ λ1 - λ= + 4
λ1= 2 and λ3 = 6

(b) The characterstic equation is

(λ - 2)(λ - 4)(λ - 6) = 0 ⇒ λ3 -12λ2 + 44λ - 48 = 0

Q.5. Find the points where the curve represented by [ x y ] Linear Algebra: Matrices - Assignment | Mathematical Methods - Physics cuts the x and y axes.
Ans. 
Linear Algebra: Matrices - Assignment | Mathematical Methods - Physics
Linear Algebra: Matrices - Assignment | Mathematical Methods - Physics
Linear Algebra: Matrices - Assignment | Mathematical Methods - Physics
⇒ x (5x - 7 y) + y (7 x + 3 y ) = 30
5x2 + 3 y2 + ( 7 xy -7 xy) = 30
Linear Algebra: Matrices - Assignment | Mathematical Methods - Physics
This is an equation of ellipse. Its graph is shown in the figure.

Q.6. Given the matrix B = Linear Algebra: Matrices - Assignment | Mathematical Methods - Physics
(a) Write the characteristic polynomial.
(b) If one of the eigenvalues of the matrix is twice repeated , find all the eigenvalues.
(c) Find the eigenvector associated with each eigenvalue.

Ans. (a) Tr (B) = 3 - 5 + 2 = 0, |B| = -16
Cofactors of diagonal elements are
B11 = -4, B22 = 0, B33 = -8
Therefore, P (A) = λ3 -12λ +16
(b) Since one of the eigenvalues of the matrix is twice repeated hence P (λ) and P'(λ) have the same roots.
P'(λ) = 3λ2 -12 ⇒ P'(λ) = 0 gives λ = ±2

Putting A = 2, makes P (λ) = 0 . Therefore λ = 2 is a twice repeated root.

From the property of roots we must have λ1λ2λ3 = -16 ⇒ 2 • 2 • λ = -16 ⇒ λ = -4. Thus λ = 2 is a twice repeated root and λ = -4 is another root.

(c) Subtracting λ = 2 down the main diagonal of B gives the matrix
Linear Algebra: Matrices - Assignment | Mathematical Methods - Physics
Here we have subtracted row 3 from row 2.
x1 - x2 + x3 = 0 and x3 = 0
This system has only one free variable hence Linear Algebra: Matrices - Assignment | Mathematical Methods - Physics is an eigenvector.

Q.7. For the matrix
Linear Algebra: Matrices - Assignment | Mathematical Methods - Physics
Ans. We see that A2 = A, i.e, A is an idempotent matrix. Therefore
Linear Algebra: Matrices - Assignment | Mathematical Methods - Physics
Linear Algebra: Matrices - Assignment | Mathematical Methods - Physics
Linear Algebra: Matrices - Assignment | Mathematical Methods - Physics
Linear Algebra: Matrices - Assignment | Mathematical Methods - Physics
Linear Algebra: Matrices - Assignment | Mathematical Methods - Physics

Q.8. Given the matrix
(a) Find the eigenvalue of matrix B .
(b) Find the eigenvectors of matrix B .
(c) Diagonalize matrix B .

(d) Find the matrix Linear Algebra: Matrices - Assignment | Mathematical Methods - Physics

(e) Using Cayley-Hamilton’s theorem find the characteristic equation satisfied by matrix B .

(f) Are the eigenvectors of matrix B linearly independent? Are the eigenvectors orthogonal to each other?
Ans. (a) eigenvalves of B
Linear Algebra: Matrices - Assignment | Mathematical Methods - Physics
⇒ (1 - λ)( 2 - λ)(3 - λ) = 0

⇒ λ = 1,λ = 2 and λ = 3
(b) Eigenvectors of B
Linear Algebra: Matrices - Assignment | Mathematical Methods - Physics
x1 + x3 = x1 ⇒ x3 = 0

2x2 = x2 ⇒ x2 = 0
x2 + 3x3 = x3
Linear Algebra: Matrices - Assignment | Mathematical Methods - Physics
For λ = 2,
Linear Algebra: Matrices - Assignment | Mathematical Methods - Physics
x1 + x3 = 2x1 ⇒ x3 = x1
2x2 = 2x2

x2 + 3x3 = 2x3 x2 = -x3
Linear Algebra: Matrices - Assignment | Mathematical Methods - Physics
Linear Algebra: Matrices - Assignment | Mathematical Methods - Physics
x1 + x3 = 3x1 ⇒ x3 = 2x1

2x2 = 3x2 ⇒ x2 = 0

x2 + 3x3 = 3x3
Linear Algebra: Matrices - Assignment | Mathematical Methods - Physics
(c) We can take Linear Algebra: Matrices - Assignment | Mathematical Methods - Physicsas three eigenvectors corresponding to λ = 1,λ = 2 and λ = 3. The matrix P diagonalizing matrix B is
Linear Algebra: Matrices - Assignment | Mathematical Methods - Physics

Hence P-1AP = D, where D = Linear Algebra: Matrices - Assignment | Mathematical Methods - Physics
(d) we can write B = PDP-1
Linear Algebra: Matrices - Assignment | Mathematical Methods - Physics
Linear Algebra: Matrices - Assignment | Mathematical Methods - Physics
Linear Algebra: Matrices - Assignment | Mathematical Methods - Physics
Linear Algebra: Matrices - Assignment | Mathematical Methods - Physics
Linear Algebra: Matrices - Assignment | Mathematical Methods - Physics
(e) The characteristic polynomial of the matrix is
P(λ) = (λ-1)(λ-2)(λ-3) = λ3 -6λ2 + 11λ - 6
Hence, according to Cayley-Hamiltoris theorem matrix equation satisfied by matrix B is B3 - 6B2 + 11B - 6I = 0

(f) We know that the eigenvectors belonging to distinct eigenvalves are linearly independent . Hence the eigenvalves of B corresponding to λ = 1,λ = 2 and λ = 3 are linearly independent. Linear Algebra: Matrices - Assignment | Mathematical Methods - Physics are the eigenvectors corresponding to λ = 1,λ = 2 and λ = 3 respectively.
From orthogonality condition
Linear Algebra: Matrices - Assignment | Mathematical Methods - Physics
which is possible only when either k = 0 or a = 0. But k = 0 and a = 0 is not allowed. Linear Algebra: Matrices - Assignment | Mathematical Methods - Physics are not orthogonal. Similarly we see that no pair of eigenvectors are orthogonal.

Q.9. If A and B are two matrices such that AB and A + B are both defined. What can be said about the relation between the order of the two matrices? Are A and B square?
Ans. Let A be an m x n matrix. Since A + B is defined, therefore B should also be an m x n . Further since AB in defined, hence m = n , hence A and B are square matrices of the same order.

Q.10. Let A, B, C and D be four non -zero matrices such that A = Linear Algebra: Matrices - Assignment | Mathematical Methods - Physics
Linear Algebra: Matrices - Assignment | Mathematical Methods - Physics If AB = CD , find matrix B.
Ans. 
Linear Algebra: Matrices - Assignment | Mathematical Methods - Physics

Since AB = CD ⇒ AB = I

Hence matrix B is the inverse of matrix A.
Linear Algebra: Matrices - Assignment | Mathematical Methods - Physics
Linear Algebra: Matrices - Assignment | Mathematical Methods - Physics

Q.11. If A is a matrix such that A = Linear Algebra: Matrices - Assignment | Mathematical Methods - Physicsthen
(a) Find matrix A100 + A99 + A98 + A97 + A2 + A .
(b) Find the Trace of matrix in part (a).
(c) Find the Determinant of matrix in part (a).

Ans. (a)
Linear Algebra: Matrices - Assignment | Mathematical Methods - Physics
⇒ A= AA2 = A2 = A, A4 = A2 = AA5 = A2 = A....
Thus we have see that all powers of A are equation to A, hence we can write
A100 + A99 + A98 + A97 + ••• A2 + A
= A + A + A + ••• upto hundred times
Linear Algebra: Matrices - Assignment | Mathematical Methods - Physics

(b)    The Trace of A100 + A99 + A98 + A97 + ••• A2 + A = 200+300 - 300 = 200

(c)    The determinant of A100 + A99 + A98 + ••• A2 + A is
Linear Algebra: Matrices - Assignment | Mathematical Methods - Physics
= 106 [2 (-9 [ 8) + 2 (3 - 4)-4 (2 - 3)] = 0

Q.12. If A and B are two matrices such that AB = A and BA = B . If A2 + B2 = 2α (A + B) then find the value of α .
Ans. We have AB = A

⇒ A (BA) = A[∴B = BA]

AB)A = A = A2 = A    (i)
Again BA = B ⇒ BAB = B (Since A = AB)

(BA)B = B ⇒ B2 = B    (ii)
Adding equations (i) and (ii)

A2 + B2 = A + B    (iii)
From the question
A2 + B2 = 2α (A + B)    (iv)
From (iii) and (iv) Linear Algebra: Matrices - Assignment | Mathematical Methods - Physics

Q.13. If Linear Algebra: Matrices - Assignment | Mathematical Methods - Physics
(a) Find Tr (A) + Tr (B ) + Tr (C ).

(b) Find Tr (AT ) Tr (B).Tr (CT ).

(c) Find Tr (A-1 ).Tr ( B-1) Tr (C-1).

(d) Find the product of determinant of three matrices.
Ans. |A| = (8 - 0)-2 (0 - 0) + 3 (0 - 0) = 8

|B| = 2 (12 - 0) = 24

|C| =10

Since the eigenvalues of a triangular or diagonal matrix are just the diagonal elements hence

(a) Tr ( A) + Tr (B) + Tr (C ) = 7 + 9 + 8 = 24

(b) Tr (AT) - Tr (BT)• Tr (CT ) = 7 • 9 • 8 = 504
(c) Tr (A-1)• Tr (B-1)• Tr (C-1) = Linear Algebra: Matrices - Assignment | Mathematical Methods - Physics
Linear Algebra: Matrices - Assignment | Mathematical Methods - Physics
(d) det ( A)• det ( B )• det (C)

=8 x 24 x 10
= 1920

Q.14. Given the matrices A = Linear Algebra: Matrices - Assignment | Mathematical Methods - Physics

(a) Calculate B-1 AB and B-1A-1B.

(b) What is the relation between the two matrices of part (a)
Ans. Linear Algebra: Matrices - Assignment | Mathematical Methods - Physics
|A| = [0 - 2] +1[0 + 4] + [-8 - 0] = -6
Linear Algebra: Matrices - Assignment | Mathematical Methods - Physics
|B| = 1 + 0 + 1 x 0 = 1
Linear Algebra: Matrices - Assignment | Mathematical Methods - Physics
(a) Linear Algebra: Matrices - Assignment | Mathematical Methods - Physics
Linear Algebra: Matrices - Assignment | Mathematical Methods - Physics
Linear Algebra: Matrices - Assignment | Mathematical Methods - Physics
And, D = B-1 A-1B =Linear Algebra: Matrices - Assignment | Mathematical Methods - Physics
Linear Algebra: Matrices - Assignment | Mathematical Methods - Physics
Linear Algebra: Matrices - Assignment | Mathematical Methods - Physics
Linear Algebra: Matrices - Assignment | Mathematical Methods - Physics

Q.15. Let Linear Algebra: Matrices - Assignment | Mathematical Methods - Physics.If A = 3 is one of the eigenvalues of the matrix then

(a) Find other eigenvalues of the matrix.
(b) To each of the eigenvalues of the matrix find the associated eigenvectors.

Ans. (a) The characteristic polynomial of the given matrix is

P(λ) = λ3 -Tr(A)λ2 +(A11 + A22 + A33)λ - Linear Algebra: Matrices - Assignment | Mathematical Methods - Physics
Where Tr (A) is the trace of matrix A. A11, A22 and A33 are cofactors of diagonal elements andLinear Algebra: Matrices - Assignment | Mathematical Methods - Physics is the determinant of matrix A .
Here, Tr(A) = 4 + 5 + 2 = 11 and |A| = 4(l2)-1(6)-1(-3) = 45
Linear Algebra: Matrices - Assignment | Mathematical Methods - Physics
⇒ P (2) = λ3-11λ2 + 39λ-45
Since A = 3 is a root of this polynomial hence (A-3) is a factor of P(λ). Dividing P(λ) by A-3 we obtain

P(λ) = (λ-3)(λ2 -8λ +15) = (λ-3)(λ-3)(λ-5)
Thus, λ = 3 is a repeated eigenvalue. And λ = 3 is another eigenvalue.
(b) Subtracting λ = 3 down the diagonal of matrix A we obtain the matrix
Linear Algebra: Matrices - Assignment | Mathematical Methods - Physics
Hence we can take (1,-1,0) and (1,0,1) as two linearly independent eigenvalues. Subtracting λ = 5 down the matrix diagonal gives the matrix.
Linear Algebra: Matrices - Assignment | Mathematical Methods - Physics
Thus, x1 - x3 = 0

X2 - 2 x3 = 0
Here we have only one free variable, hence (1,2,1) is an eigenvector corresponding to A = 5.

Q.16. For the matrix
Linear Algebra: Matrices - Assignment | Mathematical Methods - Physics

(a) Find the eigenvalues and eigenvectors .

(b) Find the matrix P which digonalizes matrix A .

(c) Write the diagonal form of A .

(d) Find the Trace and determinant of matrix e2A .
Ans. (a) Given Linear Algebra: Matrices - Assignment | Mathematical Methods - Physics
Solving above matrix for eigenvalues:
Linear Algebra: Matrices - Assignment | Mathematical Methods - Physics

⇒ (2 - λ)(1 - λ)(3 - λ) = 0

λ = 1, λ2 = 3, λ3 = 2
Eigenvectors corresponding to respective eigenvalues, on solving we get as:
Linear Algebra: Matrices - Assignment | Mathematical Methods - Physics
(b) Since we have three linearly independent eigenvectors, we can digonalise matrix A . Putting the eigenvectors of A as column vectors , we obtain
Linear Algebra: Matrices - Assignment | Mathematical Methods - Physics
(c) Now, inverse of P is
Linear Algebra: Matrices - Assignment | Mathematical Methods - Physics
Thus, Diagonalised matrix of given matrix A is
D = P-1AP
Linear Algebra: Matrices - Assignment | Mathematical Methods - Physics
(d) The eigenvalues of matrix e2A are e2,e4, and e6.
Hence the trace of e2A is e2 (1 + e2 + e4)
The determinant of e2A is e2.e4.e6 = e12

Q.17. For the Pauli’s spin matrices
Linear Algebra: Matrices - Assignment | Mathematical Methods - Physics

(a) Find the matrix Linear Algebra: Matrices - Assignment | Mathematical Methods - Physics

(b) Find the inverse of each of Pauli’s spin matrices.

(c) Find the eigenvalves of each Pauli’s spin matrix.

(d) Find the normalized Eigenvectors of each Pauli’s spin matrix.

(e) Diagonalize each of the Pauli’s matrices.

(f) Find the trace and determinant of matrix Linear Algebra: Matrices - Assignment | Mathematical Methods - Physics

(g) Find the matrices Linear Algebra: Matrices - Assignment | Mathematical Methods - PhysicsandLinear Algebra: Matrices - Assignment | Mathematical Methods - Physics

(h) Find the trace and determinant of matrix Linear Algebra: Matrices - Assignment | Mathematical Methods - Physics
Ans. (a) Linear Algebra: Matrices - Assignment | Mathematical Methods - Physics
Linear Algebra: Matrices - Assignment | Mathematical Methods - Physics
Linear Algebra: Matrices - Assignment | Mathematical Methods - Physics

(b) From the above relationLinear Algebra: Matrices - Assignment | Mathematical Methods - Physics we see that each Pauli’s spin matrix its

own inverse. A matrix which is its own inverse is called involutory. Thus each Pauli’s spin matrix is involutory.
Linear Algebra: Matrices - Assignment | Mathematical Methods - Physics
(c) Eigenvalues of Linear Algebra: Matrices - Assignment | Mathematical Methods - Physics
Linear Algebra: Matrices - Assignment | Mathematical Methods - Physics
Eigenvalues of Linear Algebra: Matrices - Assignment | Mathematical Methods - Physics
Eigenvalues of Linear Algebra: Matrices - Assignment | Mathematical Methods - Physics
We see that the Eigenvalues of each Pauli’s spin matrices are ±1.
(d) Eigenvectors of σ1:
For λ = 1, we have , Linear Algebra: Matrices - Assignment | Mathematical Methods - Physics
For λ = -1 ,we have, Linear Algebra: Matrices - Assignment | Mathematical Methods - Physics
Eigen vectors of σ2
For λ = 1, we have , Linear Algebra: Matrices - Assignment | Mathematical Methods - Physics
For λ = -1, we have , Linear Algebra: Matrices - Assignment | Mathematical Methods - Physics
Eigen vectors of σ3 are, Linear Algebra: Matrices - Assignment | Mathematical Methods - Physics
(e) Diagonalisation of σ1,
Linear Algebra: Matrices - Assignment | Mathematical Methods - Physics
Linear Algebra: Matrices - Assignment | Mathematical Methods - Physics
Linear Algebra: Matrices - Assignment | Mathematical Methods - Physics
Diagonalisation of σ2,
Linear Algebra: Matrices - Assignment | Mathematical Methods - Physics
Linear Algebra: Matrices - Assignment | Mathematical Methods - Physics
Linear Algebra: Matrices - Assignment | Mathematical Methods - Physics
σ3 is already diagonalised.
Linear Algebra: Matrices - Assignment | Mathematical Methods - Physics
Linear Algebra: Matrices - Assignment | Mathematical Methods - Physics
Linear Algebra: Matrices - Assignment | Mathematical Methods - Physics
Linear Algebra: Matrices - Assignment | Mathematical Methods - Physics
Linear Algebra: Matrices - Assignment | Mathematical Methods - Physics
Linear Algebra: Matrices - Assignment | Mathematical Methods - Physics
Linear Algebra: Matrices - Assignment | Mathematical Methods - Physics
Linear Algebra: Matrices - Assignment | Mathematical Methods - Physics
(g) Linear Algebra: Matrices - Assignment | Mathematical Methods - Physics
Linear Algebra: Matrices - Assignment | Mathematical Methods - Physics
Linear Algebra: Matrices - Assignment | Mathematical Methods - Physics
Linear Algebra: Matrices - Assignment | Mathematical Methods - Physics
Linear Algebra: Matrices - Assignment | Mathematical Methods - Physics
Linear Algebra: Matrices - Assignment | Mathematical Methods - Physics

Linear Algebra: Matrices - Assignment | Mathematical Methods - Physics
Now, Linear Algebra: Matrices - Assignment | Mathematical Methods - PhysicsSince σ3 is diagonal we obtainLinear Algebra: Matrices - Assignment | Mathematical Methods - Physics

(h) σ1σ2σ3

As, σ1σ2 = iσ3 ⇒ σ1σ2σ3 = Linear Algebra: Matrices - Assignment | Mathematical Methods - Physics
Linear Algebra: Matrices - Assignment | Mathematical Methods - Physics
Trace λ1 + λ2 = i + i = 2i

The document Linear Algebra: Matrices - Assignment | Mathematical Methods - Physics is a part of the Physics Course Mathematical Methods.
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1. What is linear algebra and why is it important in mathematics?
Ans. Linear algebra is a branch of mathematics that deals with vectors, vector spaces, linear transformations, and systems of linear equations. It is important in mathematics because it provides a framework for solving problems involving linear relationships, which are fundamental in many areas such as physics, computer science, economics, and engineering.
2. What is a matrix and how is it used in linear algebra?
Ans. A matrix is a rectangular array of numbers or symbols arranged in rows and columns. In linear algebra, matrices are used to represent and operate on linear transformations and systems of linear equations. They are also used to represent data in various fields such as statistics, computer graphics, and network analysis.
3. How do you multiply two matrices together?
Ans. To multiply two matrices together, the number of columns in the first matrix must be equal to the number of rows in the second matrix. Each element in the resulting matrix is calculated by taking the dot product of the corresponding row from the first matrix and the corresponding column from the second matrix. The process is repeated for all rows and columns to obtain the final matrix product.
4. What is the determinant of a matrix and what does it signify?
Ans. The determinant of a square matrix is a scalar value that can be calculated using a specific formula. It signifies several important properties of the matrix, such as whether the matrix is invertible (non-zero determinant) or singular (zero determinant). It also provides information about the scaling factor of linear transformations represented by the matrix.
5. How can linear algebra be applied in real-world problems?
Ans. Linear algebra has numerous applications in real-world problems. It can be used to solve systems of linear equations, analyze networks and circuits, perform data analysis and dimensionality reduction, solve optimization problems, and model physical phenomena. It is widely used in fields such as physics, engineering, computer science, economics, and data science.
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