Matrix

TRACE OF A MATRIX

The sum of the elements of a square matrix A lying along the principal diagonal is called the trace of A i.e. tr(A)

Thus if A = [a_ij]_n×n Then tr(A) = ∑_i=1ⁿ a_ii = a₁₁ + a₂₂ + ... + a_nn

Illustration 1:

The trace of the matrix A = 1 3 4, 2 -1 6, 3 1 4 is

(A) 2

(B) 4

(C) -2

(D) 1

Solution: tr (A) = 1 + (-1) + 4 = 4.

Diagonal Matrix

A square matrix all of whose elements except those in the leading diagonal, are zero is called a diagonal matrix. For a square matrix A = [a_ij]_n×n to be a diagonal matrix, a_ij = 0, whenever i ≠ j.

Scalar Matrix

A diagonal matrix whose all the leading diagonal elements are equal is called a scalar matrix.

For a square matrix A = [a_ij]_n×n to be a scalar matrix a_ij = {0, i ≠ j; m, i = j}, where m ≠ 0.

Unit Matrix or Identity Matrix

A diagonal matrix of order n which has unity for all its diagonal elements, is called a unit matrix of order n and is denoted by I_n.

Thus a square matrix A = [a_ij]_n×n is a unit matrix if a_ij = {1, i = j; 0, i ≠ j}

Triangular Matrix

A square matrix in which all the elements below the diagonal elements are zero is called Upper Triangular matrix and a square matrix in which all the elements above diagonal elements are zero is called Lower Triangular matrix.

Given a square matrix A = [a_ij]_n×n, For upper triangular matrix, a_ij = 0, i > j and for lower triangular matrix, a_ij = 0, i < j

Transpose of a Matrix

The matrix obtained from any given matrix A, by interchanging rows and columns, is called the transpose of A and is denoted by A^T.

If A = [a_ij]_m×n and A^T = [b_ij]_n×m then b_ij = a_ji, ∀ i, j.

Conjugate of a Matrix

The matrix obtained from any given matrix A containing complex number as its elements, on replacing its elements by the corresponding conjugate complex numbers is called conjugate of A and is denoted by Ā.

Transpose Conjugate of a Matrix

The transpose of the conjugate of a matrix A is called transposed conjugate of A and is denoted by A^θ. The conjugate of the transpose of A is the same as the transpose of the conjugate of A i.e.

(A^T)‾ = (Ā)^T = A^θ. If A = [a_ij]_m×n, then A^θ = [b_ji]_n×m where b_ji = ā_ij i.e. the (j, i)^th element of A^θ = the conjugate of (i, j)^th element of A.

Illustration:

If A = 1 2

3 0 and B = 3 4 1 6 then (AB)^T is

(A) 5 9 16 12

(B) 5 9, -16 12

(C) 5 9, 16 12

(D) none of these

Solution: (C) AB = 1×3+2×1 1×4+2×6 3×3+0×1 3×4+0×6 = 5 16 9 12 (AB)^T = 5 9 16 12

Do Check: Differential Equation

ALGEBRA OF MATRICES

Illustration:

If A = 2 3 1

1 3 2 and B = 1 2 2 1 1 3, AB is (A) equal to BA

(B) not equal to BA

(C) unique matrix

(D) none of these

Solution: (B) A·B = 2 3 1 1 3 2 · 1 2 2 1 1 3 = 2+6+1 4+3+3 1+6+2 2+3+6 = 9 10 9 11 B·A = 1 2 2 1 1 3 · 2 3 1 1 3 2 = 2+2 3+6 1+4 4+1 6+3 2+2 2+3 3+9 1+6 = 4 9 5 5 9 4 5 12 7 Thus A·B ≠ B·A.

Exercise 1: (i) If A = 1 -2 4

2 3 2 3 1 5 and B = 0 -2 4 1 3 2 -1 1 5, then A + B is (A) 1 -2 4 3 3 2 2 1 5 (B) 1 -2 8 3 3 4 2 1 10 (C) 1 -4 8 3 6 4 2 2 10 (D) none of these(ii) If A² = 8A + KI, where A = 1 0 -1 7, then k is (A) 7 (B) -7 (C) 1 (D) -1

SPECIAL MATRICES

Symmetric and Skew Symmetric Matrices

A square matrix A = [a_ij] is said to be symmetric when a_ij = a_ji for all i and j, i.e. A = A^T. If a_ij = -a_ji for all i and j and all the leading diagonal elements are zero, then the matrix is called a skew symmetric matrix, i.e. A = -A^T.

Singular and Non-singular Matrix

Any square matrix A is said to be non-singular if |A| ≠ 0, and a square matrix A is said to be singular if |A| = 0.

Do Check: Statics and Dynamics

Unitary Matrix

A square matrix is said to be unitary if A^θA = I since |A^θ| = |A| and |A^θ||A| = |A^θA| therefore if A^θA = I, we have |A^θ||A| = 1.

Hermitian and Skew-Hermitian Matrix

A square matrix A = [a_ij] is said to be Hermitian matrix if ā_ij = a_ji ∀ i, j i.e. A = A^θ and a square matrix, A = [a_ij] is said to be a skew-Hermitian if ā_ij = -a_ji, ∀ i, j. i.e. A = -A^θ.

For example: 0 -2+i 2-i 0, 3i -3+2i -1-i 3-2i -2i -2-4i 1+i 2+4i 0 are skew-Hermitian matrices.a b+ic b-ic d, 3 3-4i 5+2i 3+4i 5 2-i 5-2i 2+i 2 are Hermitian matrices.

Orthogonal Matrix

Any square matrix A of order n is said to be orthogonal if AA^T = A^TA = I_n.

Idempotent Matrix

A square matrix A is called idempotent provided it satisfies the relation A² = A.

Involutary Matrix

A square matrix A is said to be involutary if A² = I.

Nilpotent Matrix

A square matrix A is called a nilpotent matrix if there exists a positive integer m such that A^m = O, where O is a null matrix. If m is the least positive integer such that A^m = O, then m is called the index of the nilpotent matrix A.

Illustration 4:

The matrix 1 1 3

5 2 6 -2 -1 -3 is nilpotent matrix of index (A) 2 (B) 3 (C) 4 (D) 5

Solution: (C) Let A = 1 1 3 5 2 6 -2 -1 -3 ⇒ A² = 0 0 0 3 3 9 -1 -1 -3 ⇒ A³ = A²·A = 0 0 0 3 3 9 -1 -1 -3 · 1 1 3 5 2 6 -2 -1 -3 = 0 0 0 0 0 0 0 0 0 ∴ A³ = 0 i.e. A^k = 0. Here k = 3. Hence A is nilpotent matrix of index 3.

Adjoint of a Square Matrix

Let A = [a_ij] be a square matrix of order n and let C_ij be cofactor of a_ij in A. Then the transpose of the matrix of cofactors of elements of A is called the adjoint of A and is denoted by adj A.

Thus, adjA = [C_ij]^T ⇒ (adj A)_ij = C_ji. where C_ij denotes the cofactor of a_ij in A. ∴ adj A = s -r -q p^T = s -q -r p.

Theorem: Let A be a square matrix of order n. Then A(adj A) = |A| I_n = (adj A)A.

Properties:

(i). adj (adj A) = |A|^n-2 A, where A is a non-singular square matrix. (ii). |adj (adj A)| = |A|^(n-1)², where A is a non-singular square matrix.

Illustration:

If A = 2 0 0

2 2 0 2 2 2, then adj (adj A) is (A) 1 0 0 16 1 1 0 1 1 1 (B) 1 0 0 8 1 1 0 1 1 1 (C) 1 0 0 4 1 1 0 1 1 1 (D) none of these

Solution: (B) adj (adj A) = |A|^3-2 A = |A|·A |A| = 2 0 0 2 2 0 2 2 2 = 8 ∴ adj (adj A) = 2 0 0 8 2 2 0 2 2 2 = 16 0 0 16 16 0 16 16 16 = 1 0 0 8 1 1 0 1 1 1.

Inverse of a Matrix

A non-singular square matrix of order n is invertible if there exists a square matrix B of the same order such that AB = I_n = BA.

The inverse of A is given by A^-1 = (1/|A|) · adj A.

Properties of Inverse of a Matrix

(i). (Reversal Law) If A and B are invertible matrices of the same order, then AB is invertible and (AB)^-1 = B^-1A^-1. (ii). If A is an invertible square matrix, then A^T is also invertible and (A^T)^-1 = (A^-1)^T. (iii). If A is a non-singular square matrix of order n, then |adjA| = |A|^n-1. (iv). The inverse of the inverse of the matrix is the original matrix itself, i.e. (A^-1)^-1 = A. (v). If A is an invertible square matrix, then adj(A^T) = (adj A)^T. (vi). If A is a non-singular matrix, then |A^-1| = |A|^-1.

Illustration 6:

The inverse of the matrix A = a b

c 1+bc/a is (A) 1+bc -b a -c a (B) 1+bc -b a c a (C) 1+bc -b a -c a (D) none of theseSolution: Given A = a b c 1+bc/a |A| = a(1 + bc/a) - bc = 1 + bc - bc = 1 ⇒ |A| ≠ 0 ⇒ A^-1 exists A^-1 = (1/|A|) adjA = (1/1) 1+bc -b -c a

ELEMENTARY TRANSFORMATIONS OR ELEMENTARY OPERATIONS OF A MATRIX

The following three operations applied on the rows (columns) of a matrix are called elementary row (column) transformations.

(i). Interchange of any two rows (columns) If i^th row (column) of a matrix is interchanged with the j^th row (column), it will be denoted by R_i ↔ R_j (C_i ↔ C_j).

(ii). Multiplying all elements of a row (column) of a matrix by a non-zero scalar. If the elements of i^th row (column) are multiplied by non-zero scalar k, it will be denoted by R_i → R_i (k) [C_i → C_i(k)] or R_i → kR_i [C_i → kC_i]

(iii). Adding to the elements of a row (column), the corresponding elements of any other row (column) multiplied by any scalar k If k times the elements of j^th row (column) are added to the corresponding elements of the i^th row (column), it will be denoted by R_i → R_i +k R_j (C_i → C_i + kC_j)

Equivalent Matrices

If a matrix B is obtained from a matrix A by one or more elementary transformations, then A and B are equivalent matrices and we write A ~ B.

An elementary transformation is called a row transformation or a column transformation according as it is applied to rows or columns.

Illustration :

The inverse of the matrix A = 3 -1 -2

2 0 1 3 5 0 is (A) -5/8 -5/4 1/8 3/8 3/4 1/8 -5/4 -3/2 -1/4 (B) -5/8 -5/4 1/8 3/8 3/4 1/8 5/4 3/2 1/4 (C) -5/8 -5/4 1/8 3/8 3/4 1/8 5/4 3/2 1/4 (D) none of these

Solution: (A) We have A = IA or 3 -1 -2 2 0 1 3 5 0 = 1 0 0 0 1 0 0 0 1A [Through step-by-step row operations...] Hence A^-1 = -5/8 -5/4 1/8 3/8 3/4 1/8 5/4 3/2 1/4

SYSTEM OF SIMULTANEOUS LINEAR EQUATIONS

Consider the following system of n linear equations in n unknowns:

a₁₁ x₁ + a₁₂ x₂ + …+ a_1n x_n = b₁ a₂₁ x₁ + a₂₂ x₂ + …+ a_2n x_n = b₂ . . . . . . . . a_n1 x₁ + a_n2 x₂ + …+ a_nn x_n = b_n This system of equation can be written in matrix form asa₁₁ a₁₂ ...a_1n a₂₁ a₂₂ ...a_2n . . . a_n1 a_n2 ...a_nnx₁ x₂ . . . x_n = b₁ b₂ . . . b_n or AX = B The n × n matrix A is called the coefficient matrix of the system of linear equations.

Homogeneous and Non-Homogeneous System of Linear Equations

A system of equations AX = B is called a homogeneous system if B = O, where O is a null matrix. Otherwise, it is called a non-homogeneous system of equations.

Solution of a System of Equations

Consider the system of equation AX = B.

A set of values of the variables x₁, x₂,...,x_n which simultaneously satisfy all the equations is called a solution of the system of equations.

Consistent System

If the system of equations has one or more solutions, then it is said to be a consistent system of equations, otherwise it is an inconsistent system of equations.

Solution of a Non-Homogeneous System of Linear Equations

There are two methods of solving a non-homogeneous system of simultaneous linear equations.

(i). Cramer's Rule is discussed in the Chapter 'Determinants' (ii). Matrix Method: Consider the equations a₁x + b₁y + c₁z = d₁ a₂x + b₂y + c₂z = d₂ a₃x + b₃y + c₃z = d₃ …(i) If A = a₁ b₁ c₁ a₂ b₂ c₂ a₃ b₃ c₃, X = x y z and D = d₁ d₂ d₃, then the equation (i) is equivalent to the matrix equation A X = D …(ii)

Solution of Homogeneous System of Linear Equations

Let AX = O be a homogeneous system of n linear equation with n unknowns. Now if A is non-singular, then the system of equations will have a unique solution i.e. trivial solution and if A is a singular, then the system of equations will have infinitely many solutions.

FORMULAE AND CONCEPTS AT A GLANCE

1. Trace of a Matrix: tr(A) = ∑_i=1ⁿ a_ii = a₁₁ + a₂₂ + ..... + a_nn

2. Properties of Transposes:

(i) (A^T)^T = A

(ii) (A + B)^T = A^T + B^T, A and B being conformable matrices

(iii) (αA)^T = αA^T, α being scalar

(iv) (AB)^T = B^TA^T, A and B being conformable for multiplication

3. Any square matrix A of order n is said to be orthogonal if AA^T = A^TA = I_n.

4. A square matrix A is called idempotent provided it satisfies the relation A² = A.

5. A matrix such that A² = I is called involutary matrix.

6. A square matrix A is called a nilpotent matrix if there exists a positive integer m such that A^m = 0. If m is the least positive integer such that A^m = 0, then m is called the index of the nilpotent matrix A.

7. The inverse of A is given by A^-1 = (1/|A|) · adj A