JEE Mathematics Matrices MCQs Set 09

Answer: (c) \( p=r,\ q=s \)

Explanation:
1. Matrices can be added or subtracted only when they have the same order.
2. So the number of rows must match (\( p=r \)) and the number of columns must match (\( q=s \)).

Teacher's Note:
1. Addition and subtraction are defined element by element, so both matrices need the same shape.
2. Multiplication has a different condition: columns of the first equal rows of the second.

Answer: (b) Scalar matrix

Explanation:
1. All off-diagonal elements are zero, so it is a diagonal matrix.
2. All diagonal elements are equal to the same constant k, which is the definition of a scalar matrix.
3. It is a unit matrix only for \( k=1 \) and a null matrix only for \( k=0 \), so the general name is scalar matrix.

Teacher's Note:
1. Every scalar matrix is diagonal, but a diagonal matrix is scalar only if all diagonal entries are equal.
2. The most specific correct name is the answer.

Answer: (a) nk

Explanation:
1. The trace is the sum of the diagonal elements.
2. There are n diagonal elements, each equal to k, so the trace is \( nk \).

Teacher's Note:
1. Trace is defined only for square matrices.
2. For the unit matrix, k = 1 and the trace is n.

Answer: (c) \( A \) and \( B \) need not be zero matrices

Explanation:
1. Matrix multiplication does not obey the zero-product rule of real numbers.
2. Counter-example: \( A=\begin{bmatrix} 1 & 0 \\ 0 & 0 \end{bmatrix},\ B=\begin{bmatrix} 0 & 0 \\ 0 & 1 \end{bmatrix} \) gives \( AB=O \) with both non-zero.

Teacher's Note:
1. The product of two non-zero matrices can be the zero matrix.
2. This also means that AB = AC does not imply B = C unless A is non-singular.

Answer: (a) \( A-KI \) and \( B-KI \) commute

Explanation:
1. \( (A-KI)(B-KI)=AB-KA-KB+K^2I \).
2. \( (B-KI)(A-KI)=BA-KB-KA+K^2I \).
3. Since \( AB=BA \), the two products are equal, so they commute.

Teacher's Note:
1. The scalar matrix KI commutes with every square matrix of order n.
2. Hence adding or subtracting multiples of the identity preserves commutativity.

Answer: (d) (a), (b) and (c) are all correct

Explanation:
1. For diagonal matrices the product multiplies corresponding diagonal entries and all off-diagonal entries stay zero, so (a) holds.
2. The sum and the square of a diagonal matrix are diagonal, so (b) and (c) hold too.
3. All three statements are correct.

Teacher's Note:
1. Diagonal matrices of the same order commute and are closed under sums and products.
2. When all options are true, choose the option that says so.

Answer: (a) non-singular

Explanation:
1. \( AB=AC\Rightarrow A(B-C)=O \).
2. If \( A^{-1} \) exists, multiplying by it gives \( B=C \).
3. So the condition holds when A is non-singular.

Teacher's Note:
1. Cancellation of a left factor is possible only for an invertible matrix.
2. A singular matrix has a non-zero vector that it sends to zero, which gives counter-examples.

Answer: (b) \( A \) and \( B \) are square matrices of the same order

Explanation:
1. For \( A+B \), both matrices must have the same order, say \( m\times n \).
2. For \( AB \), the number of columns of A must equal the number of rows of B: \( n=m \).
3. So both are square matrices of the same order.

Teacher's Note:
1. Same order plus compatibility for multiplication forces the matrices to be square.
2. Rectangular matrices of the same order can be added but not multiplied.

Answer: (c) symmetric

Explanation:
1. Compare the entries across the main diagonal: \( a_{12}=a_{21}=\omega,\ a_{13}=a_{31}=\omega^2,\ a_{23}=a_{32}=1 \).
2. So \( A^T=A \), i.e. A is symmetric.
3. (It is not idempotent: \( a_{11} \) of \( A^2 \) is \( 1+\omega^2+\omega^4=1+\omega^2+\omega=0\ne1 \).)

Teacher's Note:
1. Check symmetry first: it needs only a look at the entries.
2. Powers of omega obey 1 + omega + omega squared = 0, which is useful for ruling out other options.

Answer: (b) orthogonal

Explanation:
1. Skew-symmetry gives \( S^T=-S \).
2. \( SS^T=S(-S)=-S^2=I \).
3. So \( SS^T=I \), which defines an orthogonal matrix.

Teacher's Note:
1. Substituting S transpose = -S into S S transpose turns the given condition into the definition of orthogonality.
2. Such matrices exist only of even order, for example the rotation by 90 degrees.

Answer: (a) Hermitian

Explanation:
1. The conjugate transpose has entry \( \overline{a_{ji}} \) at position (i, j).
2. \( \overline{a_{21}}=\overline{2+3i}=2-3i=a_{12} \), \( \overline{a_{31}}=3+4i=a_{13} \), \( \overline{a_{32}}=4-5i=a_{23} \).
3. The diagonal entries are real, so \( A^*=A \): A is Hermitian.

Teacher's Note:
1. For a Hermitian matrix the diagonal entries must be real and the off-diagonal entries must be conjugates of their mirror images.
2. It is not symmetric because a12 is not equal to a21.

Answer: (a) \( AB=BA \)

Explanation:
1. \( (A-B)(A+B)=A^2+AB-BA-B^2 \).
2. Equating to \( A^2-B^2 \) gives \( AB-BA=O \), i.e. \( AB=BA \).

Teacher's Note:
1. In general (A - B)(A + B) differs from A squared - B squared by the commutator AB - BA.
2. The identity holds exactly when A and B commute.

Answer: (b) O

Explanation:
1. \( AB=(I-B)B=B-B^2 \).
2. Since B is idempotent, \( B^2=B \), so \( AB=B-B=O \).

Teacher's Note:
1. Idempotent means B squared = B.
2. Similarly A is idempotent too, and A and B annihilate each other.

Answer: (a) symmetric

Explanation:
1. If \( A^T=A \): \( (A^2)^T=(A^T)^2=A^2 \).
2. If \( A^T=-A \): \( (A^2)^T=(A^T)^2=(-A)^2=A^2 \).
3. In both cases \( A^2 \) is symmetric.

Teacher's Note:
1. The transpose of a product reverses the order, but here both factors are the same matrix.
2. A squared is symmetric whenever A is symmetric or skew-symmetric.

Answer: (a) A

Explanation:
1. Here \( A=vv^T \) with \( v=(a,b,c)^T \).
2. \( A^2=v(v^Tv)v^T=(a^2+b^2+c^2)\,vv^T=A \).

Teacher's Note:
1. Writing the matrix as a column times a row makes the product easy: the middle factor v transpose v is just a number.
2. So A is idempotent, and it is the matrix of projection onto the vector (a, b, c).

Answer: (b) a skew-symmetric matrix

Explanation:
1. \( (A^n)^T=(A^T)^n=(-A)^n=(-1)^nA^n \).
2. For odd n, \( (A^n)^T=-A^n \), so \( A^n \) is skew-symmetric.

Teacher's Note:
1. Transpose passes through powers: (A to the n) transpose is (A transpose) to the n.
2. For even n the same argument gives a symmetric matrix.

Answer: (a) a symmetric matrix

Explanation:
1. \( (A^n)^T=(-A)^n=(-1)^nA^n=A^n \) for even n.
2. So \( A^n \) is symmetric.

Teacher's Note:
1. The sign (-1) to the n is +1 when n is even.
2. Check with n = 2: the square of a skew-symmetric matrix is symmetric (and negative semi-definite for real entries).

Answer: (c) skew-symmetric matrix

Explanation:
1. \( a_{ji}=j^2-i^2=-(i^2-j^2)=-a_{ij} \), and \( a_{ii}=0 \).
2. So \( A^T=-A \): A is skew-symmetric.

Teacher's Note:
1. Swapping i and j changes the sign of i squared - j squared.
2. The diagonal of a skew-symmetric matrix is always zero.

Answer: (b) an idempotent matrix

Explanation:
1. \( (A+B)^2=A^2+AB+BA+B^2=A+O+O+B=A+B \).

Teacher's Note:
1. Idempotent means M squared = M; the cross terms vanish because AB = BA = O.
2. Orthogonal idempotents are the algebraic form of splitting a space into complementary subspaces.

Answer: (c) a zero matrix of order one

Explanation:
1. \( C^TAC \) has order \( (1\times n)(n\times n)(n\times1)=1\times1 \), a single number.
2. \( (C^TAC)^T=C^TA^TC=-C^TAC \), and a single number equal to its own negative is zero.
3. So it is the zero matrix of order one.

Teacher's Note:
1. A 1 x 1 matrix equals its own transpose.
2. This is the matrix form of the fact that x transpose A x = 0 for skew-symmetric A (the quadratic form vanishes).

Answer: (c) \( |AB|=1 \)

Explanation:
1. For an orthogonal matrix, \( |A|^2=1 \), so (a) is true.
2. For a skew-symmetric matrix of odd order, \( |B|=|B^T|=|-B|=-|B|\Rightarrow|B|=0 \), so (b) is true.
3. \( |AB|=|A||B|=0 \), so (d) is true and (c) is false.

Teacher's Note:
1. Skew-symmetric matrices of odd order always have zero determinant.
2. The product with any matrix has determinant 0 as well, so |AB| = 1 cannot hold.

Answer: (a) 0

Explanation:
1. \( B_1,B_2,B_3 \) are the cofactors of the second column \( b_1,b_2,b_3 \).
2. The sum of products of the elements of the first column with the cofactors of the second column is the determinant with two equal columns, which is \( 0 \).

Teacher's Note:
1. Elements of one line multiplied by cofactors of a different parallel line always add up to zero.
2. Only the same line gives the determinant itself.

Answer: (b) doubled

Explanation:
1. A common factor of a row can be taken out of the determinant.
2. Doubling every element of one row multiplies the determinant by 2.

Teacher's Note:
1. Multiplying one row by k multiplies the determinant by k.
2. Multiplying the whole n x n matrix by k would multiply the determinant by k to the n.

Answer: (c) O

Explanation:
1. \( A(\operatorname{adj}A)=|A|I \) for every square matrix.
2. A is singular, so \( |A|=0 \) and the product is the zero matrix O.

Teacher's Note:
1. The relation A adj A = |A| I holds for all square matrices, singular or not.
2. For a singular matrix it gives the null matrix.

Answer: (b) identical rows

Explanation:
1. Row i of \( AB \) equals (row i of A) times B.
2. If two rows of A are equal, the corresponding rows of \( AB \) are obtained by multiplying the same row vector by B, so they are equal.

Teacher's Note:
1. Each row of a product depends only on the matching row of the first matrix.
2. Identical rows in the first factor therefore give identical rows in the product.