Formula Sheet and Example - Matrices and Determinants B Com Notes | EduRev

Business Mathematics and Statistics

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B Com : Formula Sheet and Example - Matrices and Determinants B Com Notes | EduRev

The document Formula Sheet and Example - Matrices and Determinants B Com Notes | EduRev is a part of the B Com Course Business Mathematics and Statistics.
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Addition and Subtraction of Matrices

  • A + B = B + A
  • (A + B) + C = A + (B + C)
  • k (A + B) = kA + kB

Multiplication of matrices

  • AB ≠ BA
  • (AB) C = A (BC)
  • Distributive law
    A (B + C) = AB + AC
    (A + B) C = AC + BC
  • Multiplicative identity
    For a square matrix A
    AI = IA = A

Proerties of transpose of matrix

  • (AT) T = A
  • (kA)T = kAT
  • (A + B)T = AT + BT
  • (AB)T = BT AT

Symmetric and Skew Symmetric matrices

  • Symmetric Matrix - If AT = A
  • Skew - symmetric Matrix - If AT = A
    Note: In a skew matrix, diagonal elements are always 0.
  • For any square matrix A,
    (A + AT) is a symmetric matrix
    (A − AT) is a skew-symmetric matrix

Inverse of a matrix
For a square matrix A, if
AB = BA = I
Then, B is the inverse of A
i.e. B = A−1
We will find inverse of a matrix by

  • Elementary transformation
  • Using adjoint

Properties of Inverse

  1. For a matrix A,
    A1 is unique, i.e., there is only one inverse of a matrix
  2. (A −1) −1 = A
  3. (𝑘 𝐴)−1 = 1/𝑘 𝐴 −1
    Note: This is different from
    (kA)T = k AT
  4. (A -1) T = (AT)-1
  5. (A+B)-1 = A-1 + B-1
  6. (𝐴𝐵)−1 = 𝐵−1 𝐴−1

Important things to note in Determinants

  1. Determinant of Identity matrix = 1
    det (I) = 1
  2. |AT | = |A|
  3. |AB| = |A| |B|
  4. |A −1 | = 1/|𝐴|
  5. |kA| = kn |A| where n is order of matrix
  6. Similarly,
    |−A| = |−1 × A|
    = (−1)n × |A|
  7. (adj A) A = A (adj) = |A|I
  8. Deteminant of adj A
    |"adj A| = |A | 𝑛−1
    where n is the order of determinant

Number multiplied to matrix and determinant

Formula Sheet and Example - Matrices and Determinants B Com Notes | EduRev

Problems and Solution.
Problems 1. 
Let Formula Sheet and Example - Matrices and Determinants B Com Notes | EduRev Notice that A contains every integer from 1 to 9 and that the sums of each row, column, and diagonal of A are equal. Such a grid is sometimes called a magic square. Compute the determinant of A.
Solution. We compute using the first row cofactor expansion
Formula Sheet and Example - Matrices and Determinants B Com Notes | EduRev
Formula Sheet and Example - Matrices and Determinants B Com Notes | EduRev

Problems 2. Let A and B be n×n matrices, where n is an integer greater than 1. Is it true that det(A+B) = det(A) + det(B)?
If so, then give a proof. If not, then give a counterexample.
Solution.
We claim that the statement is false.
As a counterexample, consider the matrices
Formula Sheet and Example - Matrices and Determinants B Com Notes | EduRev
Then
Formula Sheet and Example - Matrices and Determinants B Com Notes | EduRev
and we have
Formula Sheet and Example - Matrices and Determinants B Com Notes | EduRev
On the other hand, the determinants of A and B are
det(A) = 0 and det(B) = 0,
and hence det(A) + det(B) = 0 ≠ 1 = det(A+B).
Therefore, the statement is false and in general we have
det(A+B) ≠ det(A) + det(B).

Problems 3. LetFormula Sheet and Example - Matrices and Determinants B Com Notes | EduRev  Find all values of x such that A is invertible.
Solution.A matrix is invertible if and only if its determinant is non-zero.
So we first calculate the determinant of the matrix A.
By the first column cofactor expansion, we have
Formula Sheet and Example - Matrices and Determinants B Com Notes | EduRev
= 2((7 + x)x − (−3)4) = 2(x+ 7x + 12)
= 2(x + 3)(x + 4).
Thus the determinant of A is zero if and only if x = −3 or x = −4.
Therefore the matrix A is invertible for all x except x = −3 and x = −4.

Problems 4. Let Formula Sheet and Example - Matrices and Determinants B Com Notes | EduRev be a 4×4 matrix. Find all values of x so that the matrix A is singular.
Solution. We use the fact that a matrix is singular if and only if the determinant of the matrix is zero. We compute the determinant of A as follows.
Formula Sheet and Example - Matrices and Determinants B Com Notes | EduRev
Formula Sheet and Example - Matrices and Determinants B Com Notes | EduRev
by the first column cofactor expansion
Formula Sheet and Example - Matrices and Determinants B Com Notes | EduRev
by the first column cofactor expansion
= −1 + x2.
Therefore we have det(A) = x2−1. Thus det(A)=0 if and only if x = ±1.
We conclude that the matrix A is singular if and only if x = ±1.

Problems 5. Find all the values of x so that the following matrix A is a singular matrix.
Formula Sheet and Example - Matrices and Determinants B Com Notes | EduRev
Solution. Note that a matrix is singular if and only if its determinant is zero.
So we compute the determinant of the matrix A as follows.
Formula Sheet and Example - Matrices and Determinants B Com Notes | EduRev
Formula Sheet and Example - Matrices and Determinants B Com Notes | EduRev
by the third row cofactor expansion
= 0+(x−2)+(3x−2x2)
= −2x2+4x−2.
Thus the determinant of A is zero if
det(A) = −2x2+4x−2=0,
equivalently,
x2−2x+1=(x−1)2=0.
Thus, the determinant of the matrix A is zero if and only if x = 1.
Hence the matrix A is singular if and only if x = 1.

Problems 6. Find the value(s) of h for which the following set of vectors
Formula Sheet and Example - Matrices and Determinants B Com Notes | EduRev is linearly independent.
Solution. Let us consider the linear combination
x1v1 + x2v2 + x3v3 = 0.     (*)
If this homogeneous system has only zero solution x1 = x2 = x3 =0, then the vectors v1,v2,v3 are linearly independent.
We reduce the augmented matrix for the system as follows.
Formula Sheet and Example - Matrices and Determinants B Com Notes | EduRev
From this, we see that the homogeneous system (*) has only the zero solution if and only if
2h+ 3h + 1 ≠ 0.
Since we have
2h2+3h+1=(2h+1)(h+1),
Formula Sheet and Example - Matrices and Determinants B Com Notes | EduRev
In summary, the vectors v1,v2,v3 are linearly independent for any h except Formula Sheet and Example - Matrices and Determinants B Com Notes | EduRev

Problems 7. Let A be a 3×3 matrix.
Let x,y,z are linearly independent 3-dimensional vectors. Suppose that we have
Formula Sheet and Example - Matrices and Determinants B Com Notes | EduRev
Then find the value of the determinant of the matrix A.
Solution. Let B be the 3×3 matrix whose columns are the vectors x,y,z, that is,
B = [xyz].
Then we have
Formula Sheet and Example - Matrices and Determinants B Com Notes | EduRev
Then we have
Formula Sheet and Example - Matrices and Determinants B Com Notes | EduRev
(If two rows are equal, then the determinant is zero. Or you may compute the determinant by the second column cofactor expansion.)
Note that the column vectors of B are linearly independent, and hence B is nonsingular matrix. Thus the det(B)≠0.

Therefore the determinant of A must be zero.

Problems 8.  Given any constants a,b,c where a≠0, find all values of x such that the matrix A is invertible if Formula Sheet and Example - Matrices and Determinants B Com Notes | EduRev
Solution.
We know that A is invertible precisely when det(A)≠0. We therefore compute, by expanding along the first row,
Formula Sheet and Example - Matrices and Determinants B Com Notes | EduRev
Thus det(A)≠0 when ax2+bx+c≠0. We know by the quadratic formula that ax2+bx+c=0 precisely when
Formula Sheet and Example - Matrices and Determinants B Com Notes | EduRev
Therefore, A is invertible so long as x satisfies both of the following inequalities:
Formula Sheet and Example - Matrices and Determinants B Com Notes | EduRev

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