The trace of a square matrix is the sum of its main diagonal entries. If A is n×n, then tr(A) = a 11 + a 22 + … + a nn . Transpose does not change the diagonal, so tr(A) = tr(A T ) .

Write the system as AX = B , where A is the coefficient matrix. If det(A) ≠ 0 , then A is invertible and the unique solution is X = A⁻¹B . If det(A) = 0 , use ranks to decide consistency: Consistent when rank(A) = rank([A|B]) . If this common rank < number of unknowns, there are infinitely many solutions. Inconsistent when rank(A) ≠ rank([A|B]) (no solution).

Write the system as AX = B , where A is the coefficient matrix. If det(A) ≠ 0 , then A is invertible and the unique solution is X = A⁻¹B . If det(A) = 0 , use ranks to decide consistency: Consistent when rank(A) = rank([A|B]) . If this common rank < number of unknowns, there are infinitely many solutions. Inconsistent when rank(A) ≠ rank([A|B]) (no solution).

CBSE Class 12 Maths Notes on Matrices – Types, Operations & Properties

Q: A scalar matrix is a diagonal square matrix whose diagonal entries are all the same non-zero number m . Example: diag(5, 5, 5) is scalar. A unit (identity) matrix is a special scalar matrix with m = 1 . It’s denoted by I n . Every identity matrix is scalar, but not every scalar matrix is the identity (unless m = 1).

A scalar matrix is a diagonal square matrix whose diagonal entries are all the same non-zero number m . Example: diag(5, 5, 5) is scalar. A unit (identity) matrix is a special scalar matrix with m = 1 . It’s denoted by I n . Every identity matrix is scalar, but not every scalar matrix is the identity (unless m = 1).

Q: A square matrix A is invertible (non-singular) iff its determinant is non-zero: det(A) ≠ 0. If det(A) = 0, it is singular and has no inverse. Example: For A = [[a, b],[c, d]], A is invertible when ad − bc ≠ 0.

A square matrix A is invertible (non-singular) iff its determinant is non-zero: det(A) ≠ 0. If det(A) = 0, it is singular and has no inverse. Example: For A = [[a, b],[c, d]], A is invertible when ad − bc ≠ 0.

Q: A square matrix A is nilpotent if there exists a positive integer m such that A m = 0 (the zero matrix). The index of A is the smallest such m . Example: A = [[0,1],[0,0]] has A² = 0, so it’s nilpotent of index 2.

A square matrix A is nilpotent if there exists a positive integer m such that A m = 0 (the zero matrix). The index of A is the smallest such m . Example: A = [[0,1],[0,0]] has A² = 0, so it’s nilpotent of index 2.

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}

Do Check: Determinants

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 2:

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 3:

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 5:

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 these

Solution: 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 7:

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 as

a₁₁ a₁₂ ...a_1n
a₂₁ a₂₂ ...a_2n
.
.
.
a_n1 a_n2 ...a_nn x₁
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.

Illustration 8:

If the systems of the equations 3x+y+2z=3, 2x–3y–z = –3, x+2y+z= 4, has a unique solution, then

(A) x = 1, y = 2, z = -1 (B) x = 1, y = 2, z = 1
(C) x = -1, y = 2, z = 1 (D) x = 1, y = -2, z = -1

Solution: (A) We can write the given equations as
AX = B …(1)

Where, A = 3 1 2
2 -3 -1
1 2 1, X = x
y
z, B = 3
-3
4

Since, |A| = 3 1 2
2 -3 -1
1 2 1 = 3(-3+2)–1(2+1)+2(4+3) = -3 –3+14= 8 ≠ 0

From (1), we have X = A^-1B …(2)

[After calculating cofactors and adjoint matrix...]

A^-1 = (1/8) -1 3 5
-3 1 7
7 -5 -11

A^-1B = (1/8) -1 3 5
-3 1 7
7 -5 -11 · 3
-3
4 = (1/8) -3-9+20
-9-3+28
21+15-44 = (1/8) 8
16
-8 = 1
2
-1

Hence, from (2) x
y
z = 1
2
-1 ⇒ x = 1, y = 2, z = -1.

Exercise 2: (i) If A = 2 0 -1
5 1 0
0 1 3, then A^-1 – A² + 6A – 11I is

(A) 11I     (B) 10I     (C) I     (D) 0

(ii) The solution of the system of equation x + y – 2z = 1; x – y + z = 0; 2x + 3y – 4z = 2

(A) x = 1/3, y = -1, z = -1/3     (B) x = 1/3, y = 0, z = -1/3
(C) x = -1/3, y = 1, z = -1/3     (D) x = 1/3, y = 0, z = 1/3

(iii) If A = 0 -4 1
2 λ -3
1 2 -1, then A^-1 exists (i.e. A is invertible) if

(A) λ ≠ 4     (B) λ ≠ 8
(C) λ = 4     (D) none of these

ANSWER TO EXERCISES

Exercise 1. (i) C (ii) B
Exercise 2: (i) D (ii) B (iii) B

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

SOLVED EXAMPLES

For what value of x, the matrix 3-x 2 2
2 4-x 1
2 4 1-x is singular

(A) x = 1, 2 (B) x = 0, 2
(C) x = 0, 1 (D) x = 0, 3.

Sol. (D). Since, the given matrix is singular
3-x 2 2
2 4-x 1
2 4 1-x = 0

R₂ + R₃ ⇒ 3-x 2 2
0 x -x
2 4 1-x = 0

3-x 2 2
x 0 1 1
2 4 1-x = 0

⇒ x {(3 – x) (1 + x – 4) – 0 + 2 (2 – 2)} = 0 ⇒ x (3 – x) (x – 3) = 0 ⇒ x = 0, 3.

If A = 2 5 3
-3 1 2
1 2 1, then A^-1 is equal to

(A) -3/4 1/4 7/4
-1/4 1/4 5/4
5/4 1/4 13/4 (B) -3/4 1/4 5/4
-1/4 1/4 1/4
7/4 5/4 13/4

(C) -3 1 7
-1 1 5
5 1 13 (D) none of these

Sol. (A). Since, |A| = 2 5 3
-3 1 2
1 2 1 = 2(1 – 4) –5 (–3 –2 ) + 3(–6 –1 ) = –6 –5 + 15 =15 – 11 = 4 ≠ 0

∴ A is a non-singular matrix. Now,
A₁₁ = -3, A₁₂ = 1, A₁₃ = -5
A₂₁ = 1, A₂₂ = -1, A₂₃ = 1
A₃₁ = 7, A₃₂ = -5, A₃₃ = 13

Let, B = -3 1 -5
1 -1 1
7 -5 13

adj (A) = B^T = -3 1 7
1 -1 -5
-5 1 13

Hence A^-1 = (adj A)/|A| = (1/4) -3 1 7
1 -1 -5
-5 1 13 = -3/4 1/4 7/4
1/4 -1/4 -5/4
-5/4 1/4 13/4

If the trace of the matrix: A = x-1 0 2 5
3 x²-2 4 1
-1 -2 x-3 1
2 0 4 x²-6 is 0, then x is equal to

(A) {–2, 3} (B) (–2, 3)
(C) {–3, 2} (D) (–3, 2)

Sol. (C). Trace of matrix is defined as ∑_i=1ⁿ a_ii = 2x² + 2x –12 = 0 ⇒ x = –3, 2

If A and B are any two square matrices of the same order, then

(A) (AB)^T = A^TB^T (B) adj(AB) = adj(A) adj(B)
(C) (AB)^T = B^TA^T (D) AB = O ⇒ A = O or B = O

Sol. (C). Multiplication of square matrices is a square matrix of same order and by property of transpose.

Sol. (A). AB = -2 3 1
-1 2 1
-6 9 4 1 3 1
2 2 1
3 0 1 = 2+6+3 -4+2+6 -6+18+12
-6+6+0 -3+4+0 -18+18+0
-2+3+1 -1+2+1 -6+9+4 = 1 0 0
0 1 0
0 0 1

; BA = 1 3 1
2 2 1
3 0 1 -2 3 1
-1 2 1
-6 9 4 = -2+6+3 -4+2+6 -6+0+6
-3+6+9 -6+4+9 -9+0+9
-6+6+0 -2+2+4 -3+0+4 = 1 0 0
0 1 0
0 0 1

∴ AB = BA

If A = 1 tanx
-tanx 1, then the value of |A^TA^-1| is

(A) cos⁴x (B) sec²x
(C) – cos⁴x (D) 1

Sol. (D). A^T = 1 -tanx
tanx 1 ; A^-1 = (1/(1+tan²x)) 1 -tanx
tanx 1 ; A^TA^-1 = cos²x -sin²x
sin²x cos²x

| A^TA^-1| = 1

If A = 1 3
3 4 and A² – KA – 5I = 0, then K is equal to

(A) 5 (B) 3
(C) 7 (D) none of these

Sol. (A). Given A² = KA – 5I = 0 ; 10 15
15 25 = K1 3
3 4 - 5 0
0 5 = 5 15
15 20

; K1 3
3 4 = 5 15
15 20 + 5 0
0 5 = 10 15
15 25 = 51 3
3 4

∴ K = 5

If 1 -tanθ
tanθ 1 1 tanθ
-tanθ 1 = a -b
b a, then

(A) a = cos 2θ, b = sin 2θ (B) a = 1, b = 1
(C) a = sin 2θ, b = cos 2θ (D) None of these

Sol. (C). 1 -tanθ
tanθ 1 1 tanθ
-tanθ 1 = 1+tan²θ -2tanθ
2tanθ 1+tan²θ = 1/(1+tan²θ) -2tanθ/(1+tan²θ)
2tanθ/(1+tan²θ) 1/(1+tan²θ)

from given relation
∴ (1/(1+tan²θ)) 1-tan²θ -2tanθ
2tanθ 1-tan²θ = a -b
b a ⇒ a = (1-tan²θ)/(1+tan²θ) = cos2θ and

b = 2tanθ/(1+tan²θ) = sin2θ

If A = 3 x-1
2x+3 x+2 is a symmetric matrix, tan A is equal to

(A) 4 (B) 3
(C) -4 (D) -3

Sol. (C). A is symmetric ⇒ A = A^T ; 3 x-1
2x+3 x+2 = 3 2x+3
x-1 x+2

⇒ x – 1 = 2x + 3 ⇒ x = -4

If A = 4 -5 -2
5 -4 2
2 2 8, then A (adj A) equals

(A) 36 -36 18
-36 36 -18
-18 18 9 (B) -36 36 18
-36 36 -18
-18 18 9

(C) 0 0 0
0 0 0
0 0 0 (D) none of these

Sol. (D). We know that A (adj A) = |A| I, so A (adj A) must be a scalar matrix.

10.

If A = 1 0
1 1, then for all natural numbers n, Aⁿ is equal to

(A) 1 0
n 1 (B) 1 0
n n

(C) 1 0
-n 1 (D) none of these

Sol. (A). A² = AA = 1 0
2 1 ; A³ = 1 0
3 1 ...... so on Hence, Aⁿ = 1 0
n 1 ∀ n ∈ ℕ

ASSIGNMENT

1. 7 1 2
9 2 1 3
4
5 + 2 4
2 is equal to

(A) 43
44     (B) 43
45     (C) 45
44     (D) none of these

2. If A = 0 0 0
0 0 0
0 1 0, then A is

(A) an invertible matrix     (B) an idempotent matrix
(C) a nilpotent matrix     (D) none of these

3. If A = 1 2 -1
-1 1 2
2 -1 1, then det (adj (adjA)) is

(A) (14)⁴     (B) (14)³
(C) (14)²     (D) 14

4. Let A be a skew-symmetric matrix of order n then

(A) |A| = 0 if n is even     (B) |A| = 0 if n is odd
(C) |A| = 0 for all n ∈ ℕ     (D) none of these

5. If A is a skew-symmetric matrix, then trace of A is

(A) 0     (B) –1
(C) 1     (D) none of these

6. If A = a b
b a, A² = α β
β α, then

(A) α = 2ab, β = a² + b²     (B) α = a² + b², β = a² - b²
(C) α = a² + b², β = 2ab     (D) α = a² + b², β = ab

ANSWERS TO ASSIGNMENT

1. A

2. C

3. A

4. B

5. A

6. C

Frequently Asked Questions

A scalar matrix is a diagonal square matrix whose diagonal entries are all the same non-zero number m. Example: diag(5, 5, 5) is scalar. A unit (identity) matrix is a special scalar matrix with m = 1. It’s denoted by I_n. Every identity matrix is scalar, but not every scalar matrix is the identity (unless m = 1).