JEE Mathematics Matrices MCQs Set 19

Answer: (b) \(A+B\)

Explanation:
1. \(A^2=A\cdot A=A(BA)=(AB)A=BA=A\).
2. \(B^2=B\cdot B=B(AB)=(BA)B=AB=B\), so \(A^2+B^2=A+B\).

Teacher's Note:
1. Use the standard properties: \( (AB)^T = B^T A^T, A adj A = |A| I \), and for a nilpotent matrix \( N (N^2 = 0) \) expand \( (I + N)^n = I \) + nN.
2. When a matrix equation is given, try to reduce it to an identity such as \( A^2 = A \) (idempotent) or \( A^2 = I \) (involutory).

Answer: (c) \(a=\pm\dfrac1{\sqrt2},\ b=\pm\dfrac1{\sqrt6},\ c=\pm\dfrac1{\sqrt3}\)

Explanation:
1. The columns must be orthogonal unit vectors (so \(A^TA=I\)). Column 1 is \((0,a,a)\): \(2a^2=1\); column 2 is \((2b,b,-b)\): \(6b^2=1\); column 3 is \((c,-c,c)\): \(3c^2=1\).
2. Their dot products are \(0\) automatically, so \(a=\pm\dfrac1{\sqrt2}\), \(b=\pm\dfrac1{\sqrt6}\), \(c=\pm\dfrac1{\sqrt3}\).

Teacher's Note:
1. Use the standard properties: \( (AB)^T = B^T A^T, A adj A = |A| I \), and for a nilpotent matrix \( N (N^2 = 0) \) expand \( (I + N)^n = I \) + nN.
2. When a matrix equation is given, try to reduce it to an identity such as \( A^2 = A \) (idempotent) or \( A^2 = I \) (involutory).

Answer: (c) \(I\)

Explanation:
1. \(BC=\begin{pmatrix}9-8&-12+12\\6-6&-8+9\end{pmatrix}=I\).
2. So \(X=I\) and \(X^n=I\).

Teacher's Note:
1. Use the standard properties: \( (AB)^T = B^T A^T, A adj A = |A| I \), and for a nilpotent matrix \( N (N^2 = 0) \) expand \( (I + N)^n = I \) + nN.
2. When a matrix equation is given, try to reduce it to an identity such as \( A^2 = A \) (idempotent) or \( A^2 = I \) (involutory).

Answer: (a) \(2\)

Explanation:
1. Taking traces: \(\operatorname{tr}A+2\operatorname{tr}B=1-3+1=-1\) and \(2\operatorname{tr}A-\operatorname{tr}B=2-1+2=3\).
2. Solving gives \(\operatorname{tr}A=1\) and \(\operatorname{tr}B=-1\), so the difference is \(2\).

Teacher's Note:
1. Use the standard properties: \( (AB)^T = B^T A^T, A adj A = |A| I \), and for a nilpotent matrix \( N (N^2 = 0) \) expand \( (I + N)^n = I \) + nN.
2. When a matrix equation is given, try to reduce it to an identity such as \( A^2 = A \) (idempotent) or \( A^2 = I \) (involutory).

Answer: (c) \(O\)

Explanation:
1. Row 1 of \(A\) times column 1 of \(B\): \(0\cdot a^2+c\cdot ab-b\cdot ac=0\); the other entries vanish in the same way.
2. In fact \(B=vv^T\) with \(v=(a,b,c)^T\) and \(Av=0\) (as \(A\) represents the cross product with \(v\)), so \(AB=O\).

Teacher's Note:
1. Use the standard properties: \( (AB)^T = B^T A^T, A adj A = |A| I \), and for a nilpotent matrix \( N (N^2 = 0) \) expand \( (I + N)^n = I \) + nN.
2. When a matrix equation is given, try to reduce it to an identity such as \( A^2 = A \) (idempotent) or \( A^2 = I \) (involutory).

Answer: (b) \(5\times5\)

Explanation:
1. An \(n\times n\) lower triangular matrix has at least \(\dfrac{n(n-1)}2\) zeros (the entries above the diagonal).
2. \(\dfrac{n(n-1)}2=10\) gives \(n=5\).

Teacher's Note:
1. Use the standard properties: \( (AB)^T = B^T A^T, A adj A = |A| I \), and for a nilpotent matrix \( N (N^2 = 0) \) expand \( (I + N)^n = I \) + nN.
2. When a matrix equation is given, try to reduce it to an identity such as \( A^2 = A \) (idempotent) or \( A^2 = I \) (involutory).

Answer: (d) \(-(3A^2+2A+5I)\)

Explanation:
1. \(A\left(3A^2+2A+5I\right)=-I\).
2. Hence \(A^{-1}=-\left(3A^2+2A+5I\right)\).

Teacher's Note:
1. Use the standard properties: \( (AB)^T = B^T A^T, A adj A = |A| I \), and for a nilpotent matrix \( N (N^2 = 0) \) expand \( (I + N)^n = I \) + nN.
2. When a matrix equation is given, try to reduce it to an identity such as \( A^2 = A \) (idempotent) or \( A^2 = I \) (involutory).

Answer: (a) \(-YX\)

Explanation:
1. \(X^T=B^TA^T+A^TB^T=BA+AB=X\) and \(Y^T=BA-AB=-Y\).
2. \((XY)^T=Y^TX^T=(-Y)X=-YX\).

Teacher's Note:
1. Use the standard properties: \( (AB)^T = B^T A^T, A adj A = |A| I \), and for a nilpotent matrix \( N (N^2 = 0) \) expand \( (I + N)^n = I \) + nN.
2. When a matrix equation is given, try to reduce it to an identity such as \( A^2 = A \) (idempotent) or \( A^2 = I \) (involutory).

Answer: (b) \(a^2I+2abA\)

Explanation:
1. \(A^2=0\).
2. \((aI+bA)^2=a^2I+2abA+b^2A^2=a^2I+2abA\).

Teacher's Note:
1. Use the standard properties: \( (AB)^T = B^T A^T, A adj A = |A| I \), and for a nilpotent matrix \( N (N^2 = 0) \) expand \( (I + N)^n = I \) + nN.
2. When a matrix equation is given, try to reduce it to an identity such as \( A^2 = A \) (idempotent) or \( A^2 = I \) (involutory).

Answer: (c) \(2\)

Explanation:
1. \(B^2=\begin{pmatrix}1+12+14&4+20+2&2+4+12\\3+15+7&12+25+1&6+5+6\\7+3+42&28+5+6&14+1+36\end{pmatrix}=\begin{pmatrix}27&26&18\\25&38&17\\52&39&51\end{pmatrix}\).
2. \(B^2-A=I\), so \(n=2\) (the entries of \(B^3\) are much larger).

Teacher's Note:
1. Use the standard properties: \( (AB)^T = B^T A^T, A adj A = |A| I \), and for a nilpotent matrix \( N (N^2 = 0) \) expand \( (I + N)^n = I \) + nN.
2. When a matrix equation is given, try to reduce it to an identity such as \( A^2 = A \) (idempotent) or \( A^2 = I \) (involutory).

Answer: (d) a null matrix

Explanation:
1. \(A^T=A\) and \(A^T=-A\) give \(A=-A\).
2. So \(2A=0\) and \(A\) is the null matrix.

Teacher's Note:
1. Use the standard properties: \( (AB)^T = B^T A^T, A adj A = |A| I \), and for a nilpotent matrix \( N (N^2 = 0) \) expand \( (I + N)^n = I \) + nN.
2. When a matrix equation is given, try to reduce it to an identity such as \( A^2 = A \) (idempotent) or \( A^2 = I \) (involutory).

Answer: (a) a null matrix

Explanation:
1. \(A^n=\begin{pmatrix}\cos n\theta&\sin n\theta\\-\sin n\theta&\cos n\theta\end{pmatrix}\), whose entries lie between \(-1\) and \(1\).
2. Dividing by \(n\) and letting \(n\to\infty\) sends every entry to \(0\).

Teacher's Note:
1. Use the standard properties: \( (AB)^T = B^T A^T, A adj A = |A| I \), and for a nilpotent matrix \( N (N^2 = 0) \) expand \( (I + N)^n = I \) + nN.
2. When a matrix equation is given, try to reduce it to an identity such as \( A^2 = A \) (idempotent) or \( A^2 = I \) (involutory).

Answer: (d) \(B'AB\) is symmetric

Explanation:
1. \((B'AB)'=B'A'(B')'=B'A'B\).
2. Since \(A'=A\), \((B'AB)'=B'AB\), so it is symmetric.

Teacher's Note:
1. Use the standard properties: \( (AB)^T = B^T A^T, A adj A = |A| I \), and for a nilpotent matrix \( N (N^2 = 0) \) expand \( (I + N)^n = I \) + nN.
2. When a matrix equation is given, try to reduce it to an identity such as \( A^2 = A \) (idempotent) or \( A^2 = I \) (involutory).

Answer: (c) \(I+(2^n-1)Z\)

Explanation:
1. Since \(Z^k=Z\) for \(k\ge1\), \((I+Z)^n=I+\left[\binom n1+\binom n2+\dots+\binom nn\right]Z\).
2. The bracket is \(2^n-1\), so \((I+Z)^n=I+(2^n-1)Z\).

Teacher's Note:
1. Use the standard properties: \( (AB)^T = B^T A^T, A adj A = |A| I \), and for a nilpotent matrix \( N (N^2 = 0) \) expand \( (I + N)^n = I \) + nN.
2. When a matrix equation is given, try to reduce it to an identity such as \( A^2 = A \) (idempotent) or \( A^2 = I \) (involutory).

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

Explanation:
1. \((A-B)(A+B)=A^2+AB-BA-B^2\).
2. Equating with \(A^2-B^2\) gives \(AB-BA=0\), i.e. \(AB=BA\).

Teacher's Note:
1. Use the standard properties: \( (AB)^T = B^T A^T, A adj A = |A| I \), and for a nilpotent matrix \( N (N^2 = 0) \) expand \( (I + N)^n = I \) + nN.
2. When a matrix equation is given, try to reduce it to an identity such as \( A^2 = A \) (idempotent) or \( A^2 = I \) (involutory).

Answer: (a) 2 only

Explanation:
1. \(A^2=\begin{pmatrix}2&-2\\-2&2\end{pmatrix}=-2A\), so statement 1 is false.
2. \(A^3=A\cdot A^2=-2A^2=4A\), so statement 2 is true.

Teacher's Note:
1. Use the standard properties: \( (AB)^T = B^T A^T, A adj A = |A| I \), and for a nilpotent matrix \( N (N^2 = 0) \) expand \( (I + N)^n = I \) + nN.
2. When a matrix equation is given, try to reduce it to an identity such as \( A^2 = A \) (idempotent) or \( A^2 = I \) (involutory).

Answer: (b) \(5\) or \(-1\)

Explanation:
1. \(A^2=\begin{pmatrix}\alpha^2&0\\\alpha+1&1\end{pmatrix}\), so \(\alpha^2=9\), \(a=0\), \(b=\alpha+1\), \(c=1\).
2. \(a+b+c=\alpha+2\), which is \(5\) for \(\alpha=3\) and \(-1\) for \(\alpha=-3\).

Teacher's Note:
1. Use the standard properties: \( (AB)^T = B^T A^T, A adj A = |A| I \), and for a nilpotent matrix \( N (N^2 = 0) \) expand \( (I + N)^n = I \) + nN.
2. When a matrix equation is given, try to reduce it to an identity such as \( A^2 = A \) (idempotent) or \( A^2 = I \) (involutory).

Answer: (b) \(2X\)

Explanation:
1. \(X^2=\begin{pmatrix}1&-8\\0&9\end{pmatrix}\).
2. \(X^2-2X+3I=\begin{pmatrix}1-2+3&-8+4\\0&9-6+3\end{pmatrix}=\begin{pmatrix}2&-4\\0&6\end{pmatrix}=2X\).

Teacher's Note:
1. Use the standard properties: \( (AB)^T = B^T A^T, A adj A = |A| I \), and for a nilpotent matrix \( N (N^2 = 0) \) expand \( (I + N)^n = I \) + nN.
2. When a matrix equation is given, try to reduce it to an identity such as \( A^2 = A \) (idempotent) or \( A^2 = I \) (involutory).

Answer: (c) \(0\)

Explanation:
1. \(A^2=0\), so \((A+I)^{50}=I+50A\) and \((A+I)^{50}-50A=I\).
2. Thus \(a=d=1\), \(b=c=0\), and every product above contains \(b\) or \(c\), so the value is \(0\).

Teacher's Note:
1. Use the standard properties: \( (AB)^T = B^T A^T, A adj A = |A| I \), and for a nilpotent matrix \( N (N^2 = 0) \) expand \( (I + N)^n = I \) + nN.
2. When a matrix equation is given, try to reduce it to an identity such as \( A^2 = A \) (idempotent) or \( A^2 = I \) (involutory).

Answer: (c) \(\dfrac15\)

Explanation:
1. The \((1,1)\) entry gives \(-2x+7x=5x=1\).
2. Check the \((3,2)\) entry: \(14x-2-4x=10x-2=0\), also \(x=\dfrac15\); all the other entries are satisfied, so \(x=\dfrac15\).

Teacher's Note:
1. Use the standard properties: \( (AB)^T = B^T A^T, A adj A = |A| I \), and for a nilpotent matrix \( N (N^2 = 0) \) expand \( (I + N)^n = I \) + nN.
2. When a matrix equation is given, try to reduce it to an identity such as \( A^2 = A \) (idempotent) or \( A^2 = I \) (involutory).

Answer: (a) Neither 1 nor 2

Explanation:
1. \((AB)^{-1}=B^{-1}A^{-1}\), which equals \(A^{-1}B^{-1}\) only when \(AB=BA\).
2. Here \(AB=\begin{pmatrix}3&5\\1&0\end{pmatrix}\) and \(BA=\begin{pmatrix}8&7\\-5&-5\end{pmatrix}\) are different, so neither statement holds.

Teacher's Note:
1. Use the standard properties: \( (AB)^T = B^T A^T, A adj A = |A| I \), and for a nilpotent matrix \( N (N^2 = 0) \) expand \( (I + N)^n = I \) + nN.
2. When a matrix equation is given, try to reduce it to an identity such as \( A^2 = A \) (idempotent) or \( A^2 = I \) (involutory).