A matrix is an array of numbers arranged in the form of rows and columns. The number of rows and columns of a matrix are known as its dimensions which is given by m x n, where m and n represent the number of rows and columns respectively. Apart from basic mathematical operations there are certain elementary operations that can be performed on matrix namely transformations. It is a special type of matrix that can be illustrate 2d and 3d transformations. Let’s have a look on different types of elementary operations.
There are two types of elementary operations of a matrix:
For example, given the matrix A below:
A =
We apply R_{1 }↔ R_{2}_{,}and obtain:
A =
For example, given the matrix A below:
A =
We apply R_{1 }↔ 3R_{1}and obtain:
A =
For example, given the matrix A below:
A =
We apply R_{2}_{ }↔ R_{2 }+ 4R_{1} and obtain:
A =
A matrix is an array of numbers arranged in the form of rows and columns. The number of rows and columns of a matrix are known as its dimensions, which is given by m x n where m and n represent the number of rows and columns respectively. The basic mathematical operations like addition, subtraction, multiplication and division can be done on matrices. In this article, we will discuss the inverse of a matrix or the invertible vertices.
A matrix A of dimension n x n is called invertible if and only if there exists another matrix B of the same dimension, such that AB = BA = I, where I is the identity matrix of the same order. Matrix B is known as the inverse of matrix A. Inverse of matrix A is symbolically represented by A^{1}. Invertible matrix is also known as a nonsingular matrix or nondegenerate matrix.
For example, matrices A and B are given below:
Now we multiply A with B and obtain an identity matrix:
Similarly, on multiplying B with A, we obtain the same identity matrix:
It can be concluded here that AB = BA = I. Hence A^{1} = B, and B is known as the inverse of A. Similarly, A can also be called an inverse of B, or B^{1} = A.
A square matrix that is not invertible is called singular or degenerate. A square matrix is called singular if and only if the value of its determinant is equal to zero. Singular matrices are unique in the sense that if the entries of a square matrix are randomly selected from any finite region on the number line or complex plane, then the probability that the matrix is singular is 0, that means, it will “rarely” be singular.
Theorem 1
If there exists an inverse of a square matrix, it is always unique.
Proof:
Let us take A to be a square matrix of order n x n. Let us assume matrices B and C to be inverses of matrix A.
Now AB = BA = I since B is the inverse of matrix A.
Similarly, AC = CA = I.
But, B = BI = B (AC) = (BA) C = IC = C
This proves B = C, or B and C are the same matrices.
Theorem 2:
If A and B are matrices of the same order and are invertible, then (AB)^{1 }= B^{1} A^{1}.
Proof:
(AB)(AB)^{1} = I (From the definition of inverse of a matrix)
A^{1} (AB)(AB)^{1} = A^{1 }I (Multiplying A^{1} on both sides)
(A^{1} A) B (AB)^{1} = A^{1 }(A^{1 }I = A^{1 })
I B (AB)^{1} = A^{1}
B (AB)^{1} = A^{1}
B^{1} B (AB)^{1} = B^{1} A^{1}
I (AB)^{1} = B^{1} A^{1}
(AB)^{1} = B^{1} A^{1}
Matrix inversion is the method of finding the other matrix, say B that satisfies the previous equation for the given invertible matrix, say A. Matrix inversion can be found using the following methods:
Applications of Invertible Matrix
For many practical applications, the solution for the system of the equation should be unique and it is necessary that the matrix involved should be invertible. Such applications are:
Invertible Matrix Example
Now, go through the solved example given below to understand the matrix which can be invertible and how to verify the relationship between matrix inverse and the identity matrix.
Example: If then show that A is invertible matrix and B is its inverse.
Solution:
Given,
Now, finding the determinant of A,
= 3(0) – 1(5)
= 0 – 5
= 5 ≠ 0
Thus, A is an invertible matrix.
We know that, if A is invertible and B is its inverse, then AB = BA = I, where I is an identity matrix.
AB = BA = I
Therefore, the matrix A is invertible and the matrix B is its inverse.
Below are the following properties hold for an invertible matrix A:
Algebra of matrix involves the operation of matrices, such as Addition, subtraction, multiplication etc.
Let us understand the operation of the matrix in a much better way
Let A and B be two matrices of same order (i.e. comparable matrices). Then A + B is defined to be
Let A & B be two matrices of same order. Then A – B is defined as A + (–B) where – B is (–1) B.
Let λ be a scalar (real or complex number) & A = [a_{ij}]_{m × n }be a matrix. Thus the product λA is defined as λA = [b_{ij}]_{m × n} where b_{ij} = λa_{ij }for all i & j.
Note : If A is a scalar matrix, then A = λI, where λ is the diagonal element.
Consider all matrices of order m × n, whose elements are from a set F (F denote Q, R or C).
Let M_{m× n} (F) denote the set of all such matrices. Then
Let A and B be two matrices such that the number of columns of A is same as number of rows of B. i.e., A = where which is the dot product of i^{th} row vector of A and j^{th} column vector of B.
Note :
1. The product AB is defined iff the number of columns of A is equal to the number of rows of B. A is called as premultiplier & B is called as post multiplier. AB is defined BA is defined.
2. In general AB BA, even when both the products are defined.
3. A(BC) = (AB) C, whenever it is defined.
Consider all square matrices of order ‘n’. Let M_{n} (F) denote the set of all square matrices of order n, (where F is Q, R or C). Then
(a) A, B ∈ M_{n} (F) ⇒ AB ∈ M_{n}(F)
(b) In general AB ≠ BA
(c) (AB) C = A(BC)
(d) I_{n}, the identity matrix of order n, is the multiplicative identity. AI_{n} = A = I_{n}A
(e) For every non singular matrix A(i.e., A ≠ 0) of Mn (F) there exist a unique (particular) matrix B ∈ Mn (F) so that AB = I_{n} = BA. In this case we say that A & B are multiplicative inverse of one another. In notations, we write B = A^{1 }or A = B^{1.}
(f) If λ is a scalar (λA) B = λ(AB) = A(λB).
Note :
1. Let A = [a_{ij}]_{m × n}. Then AI_{n} = A & I_{m }A = A, where I_{n} & I_{m} are identity matrices of order n & m respectively.
2. For a square matrix A, A^{2} denotes AA, A^{3} denotes AAA etc.
Ex.1 For the following pairs of matrices, determine the sum and difference, if they exist.
(a)
(b)
Sol. (a) Matrices A and B are 2 × 3 and confirmable for addition and subtraction.
(b) Matrix A is 2 × 2, and B is 2 × 3. Since A and B are not the same size, they are not confirmable for addition or subtraction.
Ex.2 Find the additive inverse of the matrix A =
Sol. The additive inverse of the 3 × 4 matrix A is the 3 × 4 matrix each of whose elements is the negative of the corresponding element of A. Therefore if we denote the additive inverse of A by – A, we have . Obviously A + (–A) = (–A) + A = O, where O is the null matrix of the type 3 × 4.
Ex.3 If find the matrix D such that A + B – D = 0.
Sol. We have A + B – D = 0 ⇒ (A + B) + (D) = 0 ⇒ A + B = (D) = D
Ex.4 If verify that 3(A + B) = 3A + 3B.
Sol.
∴ 3 (A + B) = 3A + 3B, i.e. the scalar multiplication of matrices distributes over the addition of matrices.
Ex.5 The set of natural numbers N is partitioned into arrays of rows and columns in the form of matrices as
and so on. Find the sum of the elements of the diagonal in M_{n}.
Sol. Let M_{n} = (a_{ij}) where i, j = 1, 2, 3,.........,n.
We first find out a_{11} for the nth matrix; which is the nth term in the series ; 1, 2, 6,......
Let S = 1 + 2 + 6 + 15 +..... + T_{n – 1} + T_{n}.
Again writing S = 1 + 2 + 6 +.... + T_{n – 1} + T_{n}
⇒ 0 = 1 + 1 + 4 + 9 +..... + (T_{n }– T_{n – 1}) – T_{n} ⇒ T_{n} = 1 + (1 + 4 + 9 +....... upto (n – 1) terms)
= 1 + (1^{ 2 }+ 2^{2} + 3^{2} + 4^{2} +..... + (n – 1)^{2})
Ex.6
Sol.
The matrix AB is of the type 3 × 3 and the matrix BA is also of the type 3 × 3. But the corresponding elements of these matrices are not equal. Hence AB≠ BA.
Ex.7 Show that for all values of p, q, r, s the matrices,
Sol.
for all values of p, q, r,s. Hence PQ = QP, for all values of p, q, r, s.
Ex.8 where k is any positive integer.
Sol. We shall prove the result by induction on k.
We shall prove the result by induction on k.
We have Thus the result is true when k = 1.
Now suppose that the result is true for any positive integer k.
where k is any positive integer.
Now we shall show that the result is true for k + 1 if it is true for k. We have
Thus the result is true for k + 1 if it is true for k. But it is true for k = 1. Hence by induction it is true for all positive integral value of k.
Ex.9 where I is the two rowed unit matrix n is a positive integer.
Sol.
= 0 ⇒ A^{3} = A^{2} . A = 0 ⇒ A^{2} = A^{3} = A^{4} =...... A^{n} = 0
Now by binomial theorem
(a I + b A)^{n} = (a I)^{n} + ^{n}C_{1}(a I)^{n – 1} b A + ^{n}C_{2} (a I)^{n – 2} (b A)^{2} +..... + ^{n}C_{n }(b A)^{n}
= a^{n} I + ^{n}C_{1} a^{n – 1 }b I A + ^{n}C_{2 }an – 2 b^{2} I A^{2} +...... + ^{n}C_{n }b_{n }A_{n}
= a^{n} I + n a^{n – 1 }b A + 0......
Ex.10 If then find the value of (n + a).
Sol. Consider
Hence n = 9 and 2007 =
⇒ 2007 = 9a + 32 · 9 = 9(a + 32)
⇒ a + 32 = 223 ⇒ a = 191
hence a + n = 200
Ex.11 Find the matrices of transformations T_{1}T_{2} and T_{2}T_{1}, when T_{1} is rotation through an angle 60º and T_{2} is the reflection in the y–axis. Also verify that T_{1}T_{2} T_{2}T_{1}.
Sol.
...(1)
It is clear from (1) and (2), T_{1}T_{2} ≠ T_{2}T_{1}
Ex.12 Find the possible square roots of the two rowed unit matrix I.
Sol.
be square root of the matrix =
Since the above matrices are equal, therefore
a^{2} + bc =1 ...(i)
ab + bd = 0 ...(ii)
ac + cd = 0 ,,,(iii)
cb + d^{2} = 0 ....(iv)
must hold simultaneously.
If a + d = 0, the above four equations hold simultaneously if d = –a and a^{2} + bc =1
Hence one possible square root of I is
where α, β, γ are any three numbers related by the condition α^{2} + βγ = 1.
If a + d ≠ 0, the above four equations hold simultaneously if b = 0, c = 0, a = 1, d = 1 or if b = 0, c = 0, a = –1, d = –1. Hence i.e. ± I are other possible square roots of I.
Ex.13 then prove that
Sol.
...(i)
...(ii)
...(iii)
Similarly it can be shown that A^{4} = 2^{3} x^{4} E, A^{5 }= 2^{4} x^{5} E ...
D. FURTHER TYPES OF MATRICES
(a) Nilpotent matrix : A square matrix A is said to be nilpotent (of order 2) if, A^{2} = O.
A square matrix is said to be nilpotent of order p, if p is the least positive integer such that A^{p }= O
(b) Idempotent matrix : A square matrix A is said to be idempotent if, A^{2} = A. eg. is an idempotent matrix.
(c) Involutory matrix : A square matrix A is said to be involutory if A^{2} = I,I being the identity matrix. eg. A = is an involutory matrix.
(d) Orthogonal matrix : A square matrix A is said to be an orthogonal matrix if A'A = I = A'A
(e) Unitary matrix : A square matrix A is said to be unitary if is the complex conjugate of A.
Ex.14 Find the number of idempotent diagonal matrices of order n.
Sol. Let A = diag (d_{1}, d_{2},....., d_{n}) be any diagonal matrix of order n.
now A^{2} = A . A =
But A is idempotent, so A^{2} = A and hence corresponding elements of A^{2} and A should be equal
∴ or d_{1} = 0, 1; d_{2} = 0, 1;.........;d_{n} = 0, 1
⇒ each of d_{1}, d_{2} ......, dn can be filled by 0 or 1 in two ways.
⇒ Total number of ways of selecting d_{1}, d_{2},......., d_{n} = 2^{n}
Hence total number of such matrices = 2^{n}.
Ex.15 Show that the matrix A = is nilpotent and find its index.
Sol.
Thus 3 is the least positive integer such that A3 = 0. Hence the matrix A is nilpotent of index 3.
Ex.16 If AB = A and BA = B then B'A' = A' and A'B' = B' and hence prove that A' and B' are idempotent.
Sol. We have AB = A ⇒ (AB)' = A' ⇒ B'A' = A'. Also BA = B ⇒ (BA)' = B' ⇒ A'B' = B'.
Now A' is idempotent if A'2 = A'. We have A'2 =A'A' = A' (B'A') = (A'B') A' = B'A' =A'.
∴ A' is idempotent.
Again B'^{2} = B'B' = B' (A'B') = (B'A') B' = A'B' = B'. ∴ B' is idempotent.
E. TRANSPOSE OF MATRIX
Let A = [a_{ij}]_{m × n}. Then the transpose of A is denoted by A'(or A^{T}) and is defined as A' = [b_{ij}]_{n × m} where b_{ij }= a_{ji} for all i & j
i.e. A' is obtained by rewriting all the rows of A as columns (or by rewriting all the columns of A as rows).
(i) For any matrix A = [aij]_{m × n}, (A')' = A
(ii) Let λ be a scalar & A be a matrix. Then (λA)' = λA'
(iii) (A + B)' = A' + B' & (A  B)' = A'  B' for two comparable matrices A and B.
(iv) (A_{1} ± A_{2} ±.... ± A_{n})' = A_{1}' ± A_{2}' ±..... ± A_{n}', where A_{j} are comparable.
(v) Let A = [a_{ij}]_{m × p} & B = [b_{ij}]_{p × n}, then (AB)' = B'A'
(vi) (A_{1} A_{2 }......... A_{n})' = An'. A_{n1}'..................± A_{2}'. A_{1}', provided the product is defined.
(vii) Symmetric & Skew–Symmetric Matrix : A square matrix A is said to be symmetric if A' = A
i.e. Let A = [a_{ij}]n. A is symmetric iff a_{ij} = a_{ij} for all i & j.
A square matrix A is said to be skew–symmetric if A' =  A
i.e. Let A = [a_{ij}]_{n}. A is skew–symmetric iff a_{ij} = –a_{ji} for all i & j.
is a symmetric matrix
is a skew–symmetric matrix.
Note :
1. In skew–symmetric matrix all the diagonal elements are zero. (a_{ij} =  a_{ij} ⇒ a_{ij} = 0)
2. For any square matrix A, A + A' is symmetric & A A' is skew  symmetric.
3. Every square matrix can be uniquely expressed as a sum of two square matrices of which one is symmetric and the other is skew–symmetric.
209 videos443 docs143 tests

1. What are elementary operations on matrices? 
2. How can elementary operations be used to find the inverse of a matrix? 
3. What is the significance of invertible matrices in linear algebra? 
4. Can all matrices be inverted? 
5. How do operations on matrices help in solving systems of equations? 
209 videos443 docs143 tests


Explore Courses for JEE exam
