Matrices

If A = [1 0 / 0 1] then A squared is

1. O
2. A
3. 2A
4. [2 0 / 0 2]

Which of the following statements is always true for matrix multiplication?

1. AB = BA
2. AB is always defined
3. Matrix multiplication is associative
4. Matrix multiplication is commutative

If A is a square matrix and O is the zero matrix of the same order, then A + O is equal to

1. O
2. I
3. A
4. A squared

If A is a square matrix, then (A-transpose)-transpose is equal to

1. A
2. -A
3. A squared
4. I

If A = [1 2 / 3 4] and B = A-transpose, then the element in the first row and second column of B is

1. 2
2. 3
3. 4
4. 1

If I is the identity matrix of order 3, then AI is equal to

1. O
2. I
3. A
4. A squared

A square matrix A is called skew-symmetric if

1. A-transpose = A
2. A-transpose = -A
3. A = O
4. A = I

A square matrix A is called symmetric if

1. A-transpose = -A
2. A = I
3. A-transpose = A
4. A squared = A

If A = [1 2 / 3 4] and B = [2 0 / 1 5], then the element in the second row and first column of the matrix AB is

1. 4
2. 10
3. 12
4. 15

If A = [2 1 / 4 3] and B = [5 2 / 1 6], then the element in the first row and second column of the matrix A + B is

1. 2
2. 3
3. 6
4. 8

If A = [3 0 / 0 3], then A is

1. Zero matrix
2. Identity matrix
3. Scalar matrix
4. Diagonal matrix only

If A = [1 2 / 3 4], then the transpose of A is

1. [1 3 / 2 4]
2. [4 3 / 2 1]
3. [2 1 / 4 3]
4. [1 2 / 3 4]

If A = [2 1 / 3 4], then the trace of matrix A is

1. 5
2. 6
3. 7
4. 10

Which of the following matrices is a square matrix?

1. [1 2 3]
2. [1 2 / 3 4]
3. [2 / 5 / 7]
4. [1 2 3 / 4 5 6]

If A is a matrix of order 3 x 2 and B is a matrix of order 2 x 4, then the order of the product AB is

1. 2 x 2
2. 3 x 4
3. 4 x 3
4. 2 x 4

Let A be an invertible matrix, which of the following is not true?

1. |A^{-1}| = |A|^{-1}
2. |A^T| = |A|
3. |A^2| = |A|²
4. none of these

The trace of the matrix A = [[1,0,2],[-1,-5,0],[3,5,8]] is

1. 6
2. 5
3. 3
4. none of these

If AB = A and BA = B and I is a unit matrix, then (A – B)² is equal to

1. A
2. B
3. I
4. none of these

Let A = [matrix], where 0 < θ < π/2. Then A + A' is equal to [See original]

1. [See original]
2. [See original]
3. [See original]
4. none of these

If A = [matrix] and I is the identity matrix of order 2, then A^8 is equal to

1. I
2. O (zero matrix)
3. [matrix - see original]
4. [matrix - see original]

If A is a matrix such that A² + A + I = 0, then the inverse of A is

1. –(A + I)
2. A + I
3. –A – I
4. A

The inverse of a skew symmetric matrix (if it exists) is

1. a symmetric matrix
2. a skew symmetric matrix
3. a diagonal matrix
4. none of these

If A = [matrix with elements involving i and j] then Σ A_{ij} is equal to [See original]

1. 0
2. 1
3. 100
4. –100

If ax + y = 0, x + ay = 0 and kx + y = 0 have a non-trivial solution, then a =

1. a = 2
2. a = 1
3. a = 0
4. none of these

If A and B are symmetric matrices, then AB – BA is a

1. symmetric matrix
2. skew symmetric matrix
3. diagonal matrix
4. null matrix

Let A = [[0,1],[-1,0]] where a ∈ R. Then A^(-1) is equal to

1. A
2. -A
3. A' (transpose)
4. none of these

If A = [matrix] is such that A² = I and k·A = [matrix], then k is equal to [See original for matrix expressions]

1. –1/2
2. –1
3. 1/2
4. none of these

If [determinant with entries involving n] = 2, then n equals [See original for determinant]

1. –1
2. 1
3. –2
4. 2

Let A = [[1,0],[0,1]] and B = [[a,b],[c,d]], B ≠ 0. Then

1. there exist more than one but finite number of B's such that AB = BA
2. there exists exactly one B such that AB = BA
3. there exist infinitely many B's such that AB = BA
4. there cannot exist any B such that AB = BA

If A and B are two matrices such that AB = B and BA = A, then A² + B² is equal to

1. 2AB
2. 2BA
3. A + B
4. AB

If A is a skew symmetric matrix, then A² is

1. symmetric
2. skew-symmetric
3. invertible
4. none of these

The matrix [See original document for the specific matrix with entries] is

1. symmetric
2. unique
3. orthogonal
4. scalar

If A is a square matrix, then AA' is a

1. skew-symmetric
2. symmetric matrix
3. diagonal matrix
4. none of these

If A and B are square matrices of size n × n such that A² – B² = (A – B)(A + B), then which of the following will be always true?

1. AB = BA
2. either of A or B is a zero matrix
3. either of A or B is an identity matrix
4. A = B

The matrix product [See original document for matrix expressions] is

1. not defined
2. equal to [–1]
3. equal to [matrix - see original]
4. not invertible

If the value of a determinant is 11, then the square of the determinant formed by its cofactor will be (where order of determinant is 3 × 3)

1. 11
2. 121
3. 1331
4. 14641

If A = [aij]_{2x2}, where a_{ij} = i + j, then A is equal to [Note: Matrix expressions are in original document as images]

1. [[1,2],[2,3]]
2. [[2,3],[3,4]]
3. [[1,2],[3,4]]
4. none of these

If A = [aij] and B = [bij] are two matrices of same order, then A and B, equal to respectively [Note: Refer to original document for matrix expressions]

1. [See image in original doc]
2. [See image in original doc]
3. [See image in original doc]
4. none of these