JEE Advanced Maths PYP Matrices: Transpose, Symmetric & Skew Symmetric

Q 1 :

Consider the matrix $P = (\begin{matrix} 2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3 \end{matrix}) .$

Let the transpose of a matrix $X$ be denoted by $X^{T}$ . Then the number of $3 \times 3$ invertible matrices $Q$ , with integer entries, such that $Q^{- 1} = Q^{T}$ and $P Q = Q P,$ is [2025]

32
8
16
24

1

2

3

4

(3)

$Q^{- 1} = Q^{T} \Rightarrow Q Q^{T} = I$ $(given)$

Hence, $Q$ is an orthogonal matrix.

Let $Q = [\begin{matrix} a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \\ a_{3} & b_{3} & c_{3} \end{matrix}]$

$P Q = Q P$

$\Rightarrow [\begin{matrix} 2 a_{1} & 2 b_{1} & 2 c_{1} \\ 2 a_{2} & 2 b_{2} & 2 c_{2} \\ 3 a_{3} & 3 b_{3} & 3 c_{3} \end{matrix}] = [\begin{matrix} 2 a_{1} & 2 b_{1} & 3 c_{1} \\ 2 a_{2} & 2 b_{2} & 3 c_{2} \\ 2 a_{3} & 2 b_{3} & 3 c_{3} \end{matrix}]$

$\Rightarrow c_{1} = 0,$ $c_{2} = 0,$ $a_{3} = 0,$ $b_{3} = 0$

So, $Q = [\begin{matrix} a_{1} & b_{1} & 0 \\ a_{2} & b_{2} & 0 \\ 0 & 0 & c_{3} \end{matrix}]$

Since $Q$ is orthogonal,

$a_{1} a_{2} + b_{1} b_{2} = 0,$ $a_{1}^{2} + b_{1}^{2} = 1,$ $a_{2}^{2} + b_{2}^{2} = 1,$ $c_{3}^{2} = 1$

$\begin{matrix} a_{1} & b_{1} & a_{2} & b_{2} & c_{3} \\ 1 & 0 & 0 & 1, - 1 & 1, - 1 \\ - 1 & 0 & 0 & 1, - 1 & 1, - 1 \\ 0 & 1 & 1, - 1 & 0 & 1, - 1 \\ 0 & - 1 & 1, - 1 & 0 & 1, - 1 \end{matrix}$

$∴$ $Total 16 matrices$

Q 2 :

If $P$ is a $3 \times 3$ matrix such that $P^{T} = 2 P + I,$ where $P^{T}$ is the transpose of $P$ and $I$ is the $3 \times 3$ identity matrix, then there exists a column matrix $X = [\begin{matrix} x \\ y \\ z \end{matrix}] \neq [\begin{matrix} 0 \\ 0 \\ 0 \end{matrix}]$ such that [2012]

$P X = [\begin{matrix} 0 \\ 0 \\ 0 \end{matrix}]$
$P X = X$
$P X = 2 X$
$P X = - X$

1

2

3

4

(4)

$P^{T} = 2 P + I$

$\Rightarrow P = 2 P^{T} + I$

$\Rightarrow P = 2 (2 P + I) + I$

$\Rightarrow P = 4 P + 3 I$

$\Rightarrow P + I = 0$

$\Rightarrow P X + X = 0$

$\Rightarrow P X = - X$

Q 3 :

If $P = [\begin{matrix} \frac{\sqrt{3}}{2} & \frac{1}{2} \\ - \frac{1}{2} & \frac{\sqrt{3}}{2} \end{matrix}]$ and $A = [\begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix}]$ and $Q = P A P^{T}$ and $X = P^{T} Q^{2005} P,$ then $X$ is equal to [2005]

$[\begin{matrix} 1 & 2005 \\ 0 & 1 \end{matrix}]$
$[\begin{matrix} 4 + 2005 \sqrt{3} & 6015 \\ - 2005 & 4 - 2005 \sqrt{3} \end{matrix}]$
$\frac{1}{4} [\begin{matrix} 2 + \sqrt{3} & 1 \\ - 1 & 2 - \sqrt{3} \end{matrix}]$
$\frac{1}{4} [\begin{matrix} 2005 & 2 - \sqrt{3} \\ 2 + \sqrt{3} & 2005 \end{matrix}]$

1

2

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4

(1)

Given, $P = [\begin{matrix} \frac{\sqrt{3}}{2} & \frac{1}{2} \\ - \frac{1}{2} & \frac{\sqrt{3}}{2} \end{matrix}],$ $A = [\begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix}],$

$Q = P A P^{T}$ and $X = P^{T} Q^{2005} P$

Now, $Q = P A P^{T}$ $\Rightarrow Q^{2} = (P A P^{T}) (P A P^{T})$

$= P A (P^{T} P) A P^{T}$ $= P A (I A) P^{T}$ $= P A^{2} P^{T}$

Proceeding in the same way, $Q^{2005} = P A^{2005} P^{T}$

Now, $A = [\begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix}] \Rightarrow A^{2} = [\begin{matrix} 1 & 2 \\ 0 & 1 \end{matrix}]$

Proceeding in the same way, $A^{2005} = [\begin{matrix} 1 & 2005 \\ 0 & 1 \end{matrix}]$

Now, $X = P^{T} Q^{2005} P = P^{T} (P A^{2005} P^{T}) P$

$= (P^{T} P) A^{2005} (P^{T} P)$ $= I A^{2005} I = A^{2005} = [\begin{matrix} 1 & 2005 \\ 0 & 1 \end{matrix}]$

Q 4 :

Let $R = {(\begin{matrix} a & 3 & b \\ c & 2 & d \\ 0 & 5 & 0 \end{matrix}) : a, b, c, d \in {0, 3, 5, 7, 11, 13, 17, 19}} .$ Then the number of invertible matrices in $R$ is [2023]

(3780)

Let us calculate when $| R |$ $= 0$

Case I: $a d = b c = 0$

If $a d = 0$ (when none of $a$ and $d$ is 0),

$Total = 8^{2} - 1 = 15 ways$

Similarly, $b c = 0 \Rightarrow 15 ways$

$∴$ $15 \times 15 = 225$ ways for $a d = b c = 0$

Case II: $a d = b c \neq 0$

Either $a = d = b = c$ or $a \neq d,$ $b \neq c$ but $a d = b c$

${}^{7}C_{1} = 7 ways$ or ${}^{7}C_{2} \times 2 \times 2 = 84 ways$

$Total = 7 + 84 = 91 ways$

$∴$ $| R |$ $= 0$ in $225 + 91 = 316$ ways

$| R |$ $\neq 0$ in $8^{4} - 316 = 3780$

Q 5 :

Let $X$ and $Y$ be two arbitrary, $3 \times 3$ , non-zero, skew-symmetric matrices and $Z$ be an arbitrary $3 \times 3$ , non-zero, symmetric matrix. Then which of the following matrices is (are) skew-symmetric? [2015]

$Y^{3} Z^{4} - Z^{4} Y^{3}$
$X^{44} + Y^{44}$
$X^{4} Z^{3} - Z^{3} X^{4}$
$X^{23} + Y^{23}$

1

2

3

4

Select one or more options

(3, 4)

$X' = - X, Y' = - Y, Z' = Z$

$(Y^{3} Z^{4} - Z^{4} Y^{3})' = (Z^{4})' (Y^{3})' - (Y^{3})' (Z^{4})'$

$= {(Z')}^{4} {(Y')}^{3} - {(Y')}^{3} {(Z')}^{4}$

$= - Z^{4} Y^{3} + Y^{3} Z^{4} = Y^{3} Z^{4} - Z^{4} Y^{3}$

Therefore, $(Y^{3} Z^{4} - Z^{4} Y^{3})$ is a symmetric matrix.

Similarly, $X^{44} + Y^{44}$ is a symmetric matrix, and $X^{4} Z^{3} - Z^{3} X^{4}$ and $X^{23} + Y^{23}$ are skew-symmetric matrices.

Q 6 :

Match the Statements/Expressions in Column I with the Statements/Expressions in Column II and indicate your answer by darkening the appropriate bubbles in the $4 \times 4$ matrix given in the ORS. [2008]

Column I Column II

(A) The minimum value of $\frac{x^{2} + 2 x + 4}{x + 2}$ is (p) 0

(B) Let $A$ and $B$ be $3 \times 3$ matrices of real numbers, where $A$ is symmetric, $B$ is skew-symmetric, and $(A + B) (A - B) = (A - B) (A + B) .$
If ${(A B)}^{T} = {(- 1)}^{k} A B,$ where ${(A B)}^{T}$ is the transpose of the matrix $A B$ , then the possible values of $k$ are (q) 1

(C) Let $a = \log_{3} \log_{5} 2 .$ An integer $k$ satisfying $1 < 2^{(- k + 3 - a)} < 2,$
must be less than (r) 2

(D) If $\sin θ = \cos ϕ,$ then the possible values of $\frac{1}{π} (θ \pm ϕ - \frac{π}{2})$ are (s) 3

A → p, r, B → q, s; C → r, s; D → r
A → r, B → q, s; C → r, s; D → p, r
A → p, r, B → r, s; C → q, s; D → r
A → q, s, B → r, s; C → p, r; D → r

1

2

3

4

(2)

A → r, B → q, s; C → r, s; D → p, r

$(A)$ Let $y = \frac{x^{2} + 2 x + 4}{x + 2}$ $\Rightarrow \frac{d y}{d x} = \frac{x^{2} + 4 x}{{(x + 2)}^{2}}$

Now put $\frac{d y}{d x} = 0$ $\Rightarrow \frac{x^{2} + 4 x}{{(x + 2)}^{2}} = 0$ $\Rightarrow x = 0, - 4$

Now, $\frac{d^{2} y}{d x^{2}} = \frac{8}{{(x + 2)}^{3}}$

At $x = 0$ , $\frac{d^{2} y}{d x^{2}} = 1 > 0$

$∴$ $y is minimum when x = 0$

$∴$ $Minimum value of y, i.e., \frac{x^{2} + 2 x + 4}{x + 2}, is 2$

$∴$ $(A) \to (r)$

$(B)$ Since $A$ is a symmetric and $B$ is a skew-symmetric matrix,

$∴$ $A^{t} = A and B^{t} = - B$ $\dots (i)$

Now, $(A + B) (A - B) = (A - B) (A + B)$

$\Rightarrow A^{2} - A B + B A - B^{2} = A^{2} + A B - B A - B^{2}$

$\Rightarrow 2 B A = 2 A B$ $\Rightarrow A B = B A$ $\dots (i i)$

Also, ${(A B)}^{t} = {(- 1)}^{k} A B$

$\Rightarrow {(B A)}^{t} = {(- 1)}^{k} A B$ $(using (ii))$

$\Rightarrow A^{t} B^{t} = {(- 1)}^{k} A B$

$\Rightarrow - A B = {(- 1)}^{k} A B$ $[using (i)]$

$\Rightarrow k should be an odd number$

$∴$ $(B) \to (q), (s)$

$(C)$ $a = \log_{3} \log_{3} 2$ $\Rightarrow \log_{3} 2 = 3^{a}$

$\Rightarrow \frac{1}{\log_{2} 3} = 3^{a}$ $\Rightarrow \log_{2} 3 = 3^{- a}$

$∴$ $3 = 2^{3^{- a}}$ $\dots (i)$

Now, $1 < 2^{(- k + 3^{- a})} < 2$ $\Rightarrow 1 < 2^{- k} 2^{3^{- a}} < 2$

$\Rightarrow 1 < 2^{- k} \cdot 3 < 2$ $(using (i))$

$\Rightarrow \frac{1}{3} < 2^{- k} < \frac{2}{3}$ $\Rightarrow \frac{3}{2} < 2^{k} < 3$ $\Rightarrow k = 1$

$∴$ $k is less than 2 and 3$

$∴$ $(C) \to (r), (s)$

$(D)$ $\sin θ = \cos ϕ$ $\Rightarrow \cos (\frac{π}{2} - θ) = \cos ϕ$

$\Rightarrow \frac{π}{2} - θ = 2 n π \pm ϕ, n \in ℤ$ $\Rightarrow θ \pm ϕ - \frac{π}{2} = - 2 n π$

$\Rightarrow \frac{1}{π} (θ \pm ϕ - \frac{π}{2}) = - 2 n$

For $n = 0, - 1$ , possible values of $\frac{1}{π} (θ \pm ϕ - \frac{π}{2})$ are 0 and 2.

$∴$ $D \to (p), (r)$

Topic Question Set

Q 1 :

Consider the matrix $P = (\begin{matrix} 2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3 \end{matrix}) .$

Let the transpose of a matrix $X$ be denoted by $X^{T}$ . Then the number of $3 \times 3$ invertible matrices $Q$ , with integer entries, such that $Q^{- 1} = Q^{T}$ and $P Q = Q P,$ is [2025]

Q 2 :

If $P$ is a $3 \times 3$ matrix such that $P^{T} = 2 P + I,$ where $P^{T}$ is the transpose of $P$ and $I$ is the $3 \times 3$ identity matrix, then there exists a column matrix $X = [\begin{matrix} x \\ y \\ z \end{matrix}] \neq [\begin{matrix} 0 \\ 0 \\ 0 \end{matrix}]$ such that [2012]

Q 3 :

If $P = [\begin{matrix} \frac{\sqrt{3}}{2} & \frac{1}{2} \\ - \frac{1}{2} & \frac{\sqrt{3}}{2} \end{matrix}]$ and $A = [\begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix}]$ and $Q = P A P^{T}$ and $X = P^{T} Q^{2005} P,$ then $X$ is equal to [2005]

Q 4 :

Let $R = {(\begin{matrix} a & 3 & b \\ c & 2 & d \\ 0 & 5 & 0 \end{matrix}) : a, b, c, d \in {0, 3, 5, 7, 11, 13, 17, 19}} .$ Then the number of invertible matrices in $R$ is [2023]

Q 5 :

Let $X$ and $Y$ be two arbitrary, $3 \times 3$ , non-zero, skew-symmetric matrices and $Z$ be an arbitrary $3 \times 3$ , non-zero, symmetric matrix. Then which of the following matrices is (are) skew-symmetric? [2015]

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	Column I		Column II
(A)	The minimum value of $\frac{x^{2} + 2 x + 4}{x + 2}$ is	(p)	0
(B)	Let $A$ and $B$ be $3 \times 3$ matrices of real numbers, where $A$ is symmetric, $B$ is skew-symmetric, and $(A + B) (A - B) = (A - B) (A + B) .$ If ${(A B)}^{T} = {(- 1)}^{k} A B,$ where ${(A B)}^{T}$ is the transpose of the matrix $A B$ , then the possible values of $k$ are	(q)	1
(C)	Let $a = \log_{3} \log_{5} 2 .$ An integer $k$ satisfying $1 < 2^{(- k + 3 - a)} < 2,$ must be less than	(r)	2
(D)	If $\sin θ = \cos ϕ,$ then the possible values of $\frac{1}{π} (θ \pm ϕ - \frac{π}{2})$ are	(s)	3

Topic Question Set

Q 1 : Consider the matrix P=(200020003). Let the transpose of a matrix X be denoted by XT. Then the number of 3×3 invertible matrices Q, with integer entries, such that Q-1=QT and PQ=QP, is [2025]

Q 2 : If P is a 3×3 matrix such that PT=2P+I, where PT is the transpose of P and I is the 3×3 identity matrix, then there exists a column matrix X=[xyz]≠[000] such that [2012]

Q 3 : If P=[3212-1232] and A=[1101] and Q=PAPT and X=PTQ2005P, then X is equal to [2005]

Q 4 : Let R={(a3bc2d050): a,b,c,d∈{0,3,5,7,11,13,17,19}}. Then the number of invertible matrices in R is [2023]

Q 5 : Let X and Y be two arbitrary, 3×3, non-zero, skew-symmetric matrices and Z be an arbitrary 3×3, non-zero, symmetric matrix. Then which of the following matrices is (are) skew-symmetric? [2015]

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Q 1 :

Consider the matrix $P = (\begin{matrix} 2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3 \end{matrix}) .$

Let the transpose of a matrix $X$ be denoted by $X^{T}$ . Then the number of $3 \times 3$ invertible matrices $Q$ , with integer entries, such that $Q^{- 1} = Q^{T}$ and $P Q = Q P,$ is [2025]

Q 2 :

If $P$ is a $3 \times 3$ matrix such that $P^{T} = 2 P + I,$ where $P^{T}$ is the transpose of $P$ and $I$ is the $3 \times 3$ identity matrix, then there exists a column matrix $X = [\begin{matrix} x \\ y \\ z \end{matrix}] \neq [\begin{matrix} 0 \\ 0 \\ 0 \end{matrix}]$ such that [2012]

Q 3 :

If $P = [\begin{matrix} \frac{\sqrt{3}}{2} & \frac{1}{2} \\ - \frac{1}{2} & \frac{\sqrt{3}}{2} \end{matrix}]$ and $A = [\begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix}]$ and $Q = P A P^{T}$ and $X = P^{T} Q^{2005} P,$ then $X$ is equal to [2005]

Q 4 :

Let $R = {(\begin{matrix} a & 3 & b \\ c & 2 & d \\ 0 & 5 & 0 \end{matrix}) : a, b, c, d \in {0, 3, 5, 7, 11, 13, 17, 19}} .$ Then the number of invertible matrices in $R$ is [2023]

Q 5 :

Let $X$ and $Y$ be two arbitrary, $3 \times 3$ , non-zero, skew-symmetric matrices and $Z$ be an arbitrary $3 \times 3$ , non-zero, symmetric matrix. Then which of the following matrices is (are) skew-symmetric? [2015]