JEE Mathematics Matrices MCQs Set 20

Answer: (b) involutory matrix

Explanation:
1. Compute \( A^2 \) row by row: row \( 1 = (25 - 24, 40 - 40, 0) = (1, 0, 0) \); row \( 2 = (-15 + 15, -24 + 25, 0) = (0, 1, 0) \); row \( 3 = (-5 + 6 - 1, -8 + 10 - 2, 1) = (0, 0, 1) \).
2. So \( A^2 = I \), which is the definition of an involutory matrix.

Teacher's Note:
1. Multiply small matrices explicitly or use algebraic identities \( (A^2 = I, A^2 = 2A - I) \) to find patterns in powers.
2. Check dimensions first: a product is defined only when the number of columns of the left matrix equals the number of rows of the right matrix.

Answer: (a) -1

Explanation:
1. Multiply \( (I - A) \) by (\( I + A + ... + A^{k-1} \)): the product telescopes to I \( - A^k = I \).
2. So I \( + A + ... + A^{k-1} \) is the inverse of \( (I - A) \), i.e. \( (I - A)^{-1} \).
3. Hence \( p = -1 \).

Teacher's Note:
1. Multiply small matrices explicitly or use algebraic identities \( (A^2 = I, A^2 = 2A - I) \) to find patterns in powers.
2. Check dimensions first: a product is defined only when the number of columns of the left matrix equals the number of rows of the right matrix.

Answer: (a) \( x=\frac{11\pm\sqrt5}{2} \)

Explanation:
1. The inverse exists exactly when \( \det A \) is not 0.
2. \( \det A = 1[(x - 1)(x - 3) - 7] - 3[(x - 3) - 2] + 2[7 - 2(x - 1)] = (x^2 - 4x - 4) - 3x + 15 + 18 - 4x = x^2 - 11x + 29 \).
3. \( x^2 - 11x + 29 = 0 \) gives x = (11 plus or minus sqrt5)/2.

Teacher's Note:
1. Multiply small matrices explicitly or use algebraic identities \( (A^2 = I, A^2 = 2A - I) \) to find patterns in powers.
2. Check dimensions first: a product is defined only when the number of columns of the left matrix equals the number of rows of the right matrix.

Answer: (a) \( (A^T)^2=A^T \)

Explanation:
1. \( A^2 = A(A) \) = (AB)A = A(BA) = AB = \( A \), so \( A \) is idempotent.
2. Taking the transpose: \( (A^T)^2 = (A^2)^T = A^T \).

Teacher's Note:
1. Multiply small matrices explicitly or use algebraic identities \( (A^2 = I, A^2 = 2A - I) \) to find patterns in powers.
2. Check dimensions first: a product is defined only when the number of columns of the left matrix equals the number of rows of the right matrix.

Answer: (b) skew-symmetric

Explanation:
1. An odd power of a skew-symmetric matrix is skew-symmetric, since \( (A^m)^T = (A^T)^m = (-A)^m = -A^m \) for odd \( m \).
2. Each term is a scalar multiple of a skew-symmetric matrix, and a sum of skew-symmetric matrices is skew-symmetric.
3. So \( B \) is skew-symmetric.

Teacher's Note:
1. Multiply small matrices explicitly or use algebraic identities \( (A^2 = I, A^2 = 2A - I) \) to find patterns in powers.
2. Check dimensions first: a product is defined only when the number of columns of the left matrix equals the number of rows of the right matrix.

Answer: (a) \( 7A^8 \)

Explanation:
1. \( A^2 \): the third row becomes \( (a + 0 - a, 0 + b - b, 1) = (0, 0, 1) \), so \( A^2 = I \).
2. Then \( A^2 + 2A^4 + 4A^6 = I + 2I + 4I = 7I \).
3. Since \( A^8 = I \) as well, \( 7I = 7A^8 \).

Teacher's Note:
1. Multiply small matrices explicitly or use algebraic identities \( (A^2 = I, A^2 = 2A - I) \) to find patterns in powers.
2. Check dimensions first: a product is defined only when the number of columns of the left matrix equals the number of rows of the right matrix.

Answer: (b) 512

Explanation:
1. A \( 3 x 3 \) matrix has 9 entries.
2. Each entry has 2 choices, so the number is \( 2^9 = 512 \).

Teacher's Note:
1. Multiply small matrices explicitly or use algebraic identities \( (A^2 = I, A^2 = 2A - I) \) to find patterns in powers.
2. Check dimensions first: a product is defined only when the number of columns of the left matrix equals the number of rows of the right matrix.

Answer: (b) 4

Explanation:
1. The order \( m x n \) must satisfy \( m n = 20 \) with \( m \) even.
2. The possible values of \( m \) are 2, 4, 10, 20, giving \( 2 x 10, 4 x 5, 10 x 2, 20 x 1 \).
3. That is 4 types.

Teacher's Note:
1. Multiply small matrices explicitly or use algebraic identities \( (A^2 = I, A^2 = 2A - I) \) to find patterns in powers.
2. Check dimensions first: a product is defined only when the number of columns of the left matrix equals the number of rows of the right matrix.

Answer: (c) There exist infinitely many B's such that AB = BA

Explanation:
1. AB = \( [[a, 2b], [3a, 4b]] \) and BA = \( [[a, 2a], [3b, 4b]] \).
2. Equality needs \( 2b = 2a \) and \( 3a = 3b \), i.e. \( a = b \).
3. Every natural number \( a = b \) works, so there are infinitely many such \( B \).

Teacher's Note:
1. Multiply small matrices explicitly or use algebraic identities \( (A^2 = I, A^2 = 2A - I) \) to find patterns in powers.
2. Check dimensions first: a product is defined only when the number of columns of the left matrix equals the number of rows of the right matrix.

Answer: (d) none of these

Explanation:
1. \( \operatorname{tr}(A) \) = sum over \( i = 1 \) to 10 of \( \omega^{2i} \).
2. Terms repeat with period 3: \( \omega^2, \omega^4 = \omega, \omega^6 = 1 \), and each block of three terms sums to \( \omega^2 + \omega + 1 = 0 \).
3. The first nine terms sum to 0, leaving the tenth term \( \omega^{20} = \omega^2 \). So \( \operatorname{tr}(A) = \omega^2 \), which is none of 0, 1, 3.

Teacher's Note:
1. Multiply small matrices explicitly or use algebraic identities \( (A^2 = I, A^2 = 2A - I) \) to find patterns in powers.
2. Check dimensions first: a product is defined only when the number of columns of the left matrix equals the number of rows of the right matrix.

Answer: (a) 1

Explanation:
1. Since BC = CB, the binomial theorem applies: \( A^{N+1} = B^{N+1} + (N+1) B^N C \) + (terms with \( C^2 \) or higher, which are 0).
2. So \( A^{N+1} = B^N [B + (N+1) C] \).
3. Therefore \( K = N \) and \( K/N = 1 \).

Teacher's Note:
1. Multiply small matrices explicitly or use algebraic identities \( (A^2 = I, A^2 = 2A - I) \) to find patterns in powers.
2. Check dimensions first: a product is defined only when the number of columns of the left matrix equals the number of rows of the right matrix.

Answer: (d) \( \begin{bmatrix}1&0\\0&1\end{bmatrix} \)

Explanation:
1. \( A^2 = [[0, -1], [1, 0]] \) times itself = \( [[-1, 0], [0, -1]] = -I \).
2. \( A^{16} = (A^2)^8 = (-I)^8 = I \).

Teacher's Note:
1. Multiply small matrices explicitly or use algebraic identities \( (A^2 = I, A^2 = 2A - I) \) to find patterns in powers.
2. Check dimensions first: a product is defined only when the number of columns of the left matrix equals the number of rows of the right matrix.

Answer: (a) \( \begin{bmatrix}6&-10\\4&26\end{bmatrix} \)

Explanation:
1. AB = \( [2(x + y) - y, 4x - (x - y)] = [2x + y, 3x + y] = [3, 2] \).
2. Subtracting: \( x = -1 \), and then \( y = 5 \), so \( A = [[4, 5], [-2, -6]] \).
3. \( A^2 = [[16 - 10, 20 - 30], [-8 + 12, -10 + 36]] = [[6, -10], [4, 26]] \).

Teacher's Note:
1. Multiply small matrices explicitly or use algebraic identities \( (A^2 = I, A^2 = 2A - I) \) to find patterns in powers.
2. Check dimensions first: a product is defined only when the number of columns of the left matrix equals the number of rows of the right matrix.

Answer: (c) \( nA-(n-1)I \)

Explanation:
1. \( A^2 = 2A - I \). Then \( A^3 = A(2A - I) = 2A^2 - A = 2(2A - I) - A = 3A - 2I \).
2. \( A^4 = A(3A - 2I) = 3A^2 - 2A = 4A - 3I \).
3. The pattern gives \( A^n \) = nA \( - (n - 1)I \) (provable by induction).

Teacher's Note:
1. Multiply small matrices explicitly or use algebraic identities \( (A^2 = I, A^2 = 2A - I) \) to find patterns in powers.
2. Check dimensions first: a product is defined only when the number of columns of the left matrix equals the number of rows of the right matrix.

Answer: (b) \( C^TC(AB)^T \)

Explanation:
1. Orders: \( A \) is \( 3 x 3, B \) is \( 3 x 2, C \) is \( 3 x 1 \).
2. \( (a) (AB)^T \) is \( 2 x 3 \) and \( C \) is \( 3 x 1 \), so the product is defined. \( (c) C^T \) is \( 1 x 3, A \) is \( 3 x 3 \) and \( B \) is \( 3 x 2 \), so it is defined. \( (d) A^T \) is \( 3 x 3, A \) is \( 3 x 3, B \) is \( 3 x 2, B^T \) is \( 2 x 3 \) and \( C \) is \( 3 x 1 \), so it is defined.
3. \( (b) C^T C \) is \( 1 x 1 \) and \( (AB)^T \) is \( 2 x 3 \), so a \( 1 x 1 \) matrix times a \( 2 x 3 \) matrix is not defined.

Teacher's Note:
1. Multiply small matrices explicitly or use algebraic identities \( (A^2 = I, A^2 = 2A - I) \) to find patterns in powers.
2. Check dimensions first: a product is defined only when the number of columns of the left matrix equals the number of rows of the right matrix.

Answer: (b) \( a_{ii}b_{ij} \)

Explanation:
1. \( c_{ij} \) = sum over \( k \) of \( a_{ik} b_{kj} \).
2. For a diagonal matrix \( A, a_{ik} \) is non-zero only when \( k = i \), so the sum has one term.
3. \( c_{ij} = a_{ii} b_{ij} \).

Teacher's Note:
1. Multiply small matrices explicitly or use algebraic identities \( (A^2 = I, A^2 = 2A - I) \) to find patterns in powers.
2. Check dimensions first: a product is defined only when the number of columns of the left matrix equals the number of rows of the right matrix.

Answer: (c) \( \begin{bmatrix}-5&-2\\2&1\end{bmatrix} \)

Explanation:
1. The determinant of the left matrix \( B \) is 3 and \( adj B = [[3, -2], [0, 1]] \), so \( B^{-1} = (1/3)[[3, -2], [0, 1]] \).
2. A = \( B^{-1} \) times the right-hand side = \( (1/3)[[3(-1) - 2(6), 0 - 6], [6, 3]] = (1/3)[[-15, -6], [6, 3]] \).
3. A = [[-5, -2], [2, 1]].

Teacher's Note:
1. Multiply small matrices explicitly or use algebraic identities \( (A^2 = I, A^2 = 2A - I) \) to find patterns in powers.
2. Check dimensions first: a product is defined only when the number of columns of the left matrix equals the number of rows of the right matrix.

Answer: (c) \( \begin{bmatrix}5&2\\1&6\\4&3\end{bmatrix} \)

Explanation:
1. A is \( 2 x 3 \) and AB is \( 2 x 2 \), so \( B \) must be \( 3 x 2 \); this leaves options \( (c) \) and \( (d) \).
2. Test \( (c) \): row 1 of \( A \) times column 1 of \( B = 25 + 6 + 4 = 35 \), and row 1 times column \( 2 = 10 + 36 + 3 = 49 \).
3. Row 2 times columns: \( 10 - 1 + 20 = 29 \) and \( 4 - 6 + 15 = 13 \). So \( B \) = option \( (c) \).

Teacher's Note:
1. Multiply small matrices explicitly or use algebraic identities \( (A^2 = I, A^2 = 2A - I) \) to find patterns in powers.
2. Check dimensions first: a product is defined only when the number of columns of the left matrix equals the number of rows of the right matrix.

Answer: (b) k = -7

Explanation:
1. \( A^2 = [[1, 0], [-1 - 7, 49]] = [[1, 0], [-8, 49]] \).
2. \( 8A \) + kI = \( [[8 + k, 0], [-8, 56 + k]] \).
3. Comparing: \( 1 = 8 + k \) gives \( k = -7 \), and \( 49 = 56 + k \) also gives \( k = -7 \).

Teacher's Note:
1. Multiply small matrices explicitly or use algebraic identities \( (A^2 = I, A^2 = 2A - I) \) to find patterns in powers.
2. Check dimensions first: a product is defined only when the number of columns of the left matrix equals the number of rows of the right matrix.

Answer: (c) \( \alpha=a^2+b^2,\beta=2ab \)

Explanation:
1. A^2 = [[a^2 + b^2, 2ab], [2ab, a^2 + b^2]].
2. Comparing entries: \( \alpha = a^2 + b^2 \) and \( \beta \) = 2ab.

Teacher's Note:
1. Multiply small matrices explicitly or use algebraic identities \( (A^2 = I, A^2 = 2A - I) \) to find patterns in powers.
2. Check dimensions first: a product is defined only when the number of columns of the left matrix equals the number of rows of the right matrix.

Answer: (c) 18

Explanation:
1. A \( 2 x 2 \) sub-matrix is obtained by choosing 2 of the 3 rows and 2 of the 4 columns.
2. Number = \( 3C2 x 4C2 = 3 x 6 = 18 \).

Teacher's Note:
1. Multiply small matrices explicitly or use algebraic identities \( (A^2 = I, A^2 = 2A - I) \) to find patterns in powers.
2. Check dimensions first: a product is defined only when the number of columns of the left matrix equals the number of rows of the right matrix.