JEE Advanced Maths PYP Matrices: Order, Types, Addition & Subtraction

Q 1 :

If $M = (\begin{matrix} \frac{5}{2} & \frac{3}{2} \\ - \frac{3}{2} & - \frac{1}{2} \end{matrix}),$ then which of the following matrices is equal to $M^{2022} ?$ [2022]

$(\begin{matrix} 3034 & 3033 \\ - 3033 & - 3032 \end{matrix})$
$(\begin{matrix} 3034 & - 3033 \\ 3033 & - 3032 \end{matrix})$
$(\begin{matrix} 3033 & - 3032 \\ - 3032 & - 3031 \end{matrix})$
$(\begin{matrix} 3032 & 3031 \\ - 3031 & - 3030 \end{matrix})$

1

2

3

4

(1)

$Given that M = (\begin{matrix} \frac{5}{2} & \frac{3}{2} \\ - \frac{3}{2} & - \frac{1}{2} \end{matrix}) \Rightarrow M^{2} = (\begin{matrix} 4 & 3 \\ - 3 & - 2 \end{matrix})$

$\Rightarrow M^{3} = (\begin{matrix} \frac{11}{2} & \frac{9}{2} \\ - \frac{9}{2} & - \frac{7}{2} \end{matrix}) \Rightarrow M^{4} = (\begin{matrix} 7 & 6 \\ - 6 & - 5 \end{matrix}) and so on$

$\Rightarrow M^{n} = (\begin{matrix} \frac{3 n}{2} + 1 & \frac{3 n}{2} \\ - \frac{3 n}{2} & - \frac{3 n}{2} + 1 \end{matrix})$

$Now, M^{n} = (\begin{matrix} \frac{3 \times 2022}{2} + 1 & \frac{3 \times 2022}{2} \\ - \frac{3 \times 2022}{2} & - \frac{3 \times 2022}{2} + 1 \end{matrix})$

$= (\begin{matrix} 3034 & 3033 \\ - 3033 & - 3032 \end{matrix})$

$∴$ $Option (1) is correct$

Q 2 :

How many $3 \times 3$ matrices $M$ with entries from ${0, 1, 2}$ are there, for which the sum of the diagonal entries of $M^{T} M$ is 5? [2017]

126
198
162
135

1

2

3

4

(2)

$Let M = [\begin{matrix} a_{1} & a_{2} & a_{3} \\ a_{4} & a_{5} & a_{6} \\ a_{7} & a_{8} & a_{9} \end{matrix}],$ $where a_{i} \in {0, 1, 2}$

$Then M^{T} M = [\begin{matrix} a_{1} & a_{4} & a_{7} \\ a_{2} & a_{5} & a_{8} \\ a_{3} & a_{6} & a_{9} \end{matrix}] [\begin{matrix} a_{1} & a_{2} & a_{3} \\ a_{4} & a_{5} & a_{6} \\ a_{7} & a_{8} & a_{9} \end{matrix}]$

$Sum of the diagonal entries in M^{T} M = 5$

$\Rightarrow (a_{1}^{2} + a_{4}^{2} + a_{7}^{2}) + (a_{2}^{2} + a_{5}^{2} + a_{8}^{2}) + (a_{3}^{2} + a_{6}^{2} + a_{9}^{2}) = 5,$

$which is possible when$

$Case I: Five$ $a_{i}' s$ $are$ 1 $and four$ $a_{i}' s$ $are zero.$

$Which can be done in$ ${}^{9}C_{4} ways$

$= \frac{9 \times 8 \times 7 \times 6}{4 \times 3 \times 2 \times 1} = 126$

$Case II: One$ $a_{i}$ $is$ $1, one$ $a_{i}$ $is 2 and the remaining seven$ $a_{i}' s$ $are zero$

$It can be done in$ ${}^{9}C_{1} \times {}^{8}C_{1} = 9 \times 8 = 72 ways$

$∴$ $Total number of ways = 126 + 72 = 198$

Q 3 :

Let $P = [\begin{matrix} 1 & 0 & 0 \\ 4 & 1 & 0 \\ 16 & 4 & 1 \end{matrix}]$ and $I$ be the identity matrix of order 3. If $Q = [q_{i j}]$ is a matrix such that $P^{50} - Q = I,$ then $\frac{q_{31} + q_{32}}{q_{21}}$ equals [2016]

52
103
201
205

1

2

3

4

(2)

$P = [\begin{matrix} 1 & 0 & 0 \\ 4 & 1 & 0 \\ 16 & 4 & 1 \end{matrix}] = I + [\begin{matrix} 0 & 0 & 0 \\ 4 & 0 & 0 \\ 16 & 4 & 0 \end{matrix}] = I + A,$

where $A = [\begin{matrix} 0 & 0 & 0 \\ 4 & 0 & 0 \\ 16 & 4 & 0 \end{matrix}] \Rightarrow A^{2} = [\begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 16 & 0 & 0 \end{matrix}]$

and $A^{3} = [\begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix}],$ $∴$ $A^{n} = 0, \forall n \geq 3$

Now, $P^{50} = {(I + A)}^{50} = {}^{50}C_{0} I^{50} + {}^{50}C_{1} I^{49} A + {}^{50}C_{2} I^{48} A^{2} + 0$

$= I + 50 A + 25 \times 49 A^{2}$

$∴$ $Q = P^{50} - I = 50 A + 25 \times 49 A^{2}$

$\Rightarrow q_{21} = 50 \times 4 = 200,$ $q_{31} = 50 \times 16 + 25 \times 49 \times 16 = 20400,$

and $q_{32} = 50 \times 4 = 200$

$∴$ $\frac{q_{31} + q_{32}}{q_{21}} = \frac{20600}{200} = 103$

Q 4 :

Let $β$ be a real number. Consider the matrix $A = (\begin{matrix} β & 0 & 1 \\ 2 & 1 & - 2 \\ 3 & 1 & - 2 \end{matrix}) .$

If $A^{7} - (β - 1) A^{6} - β A^{5}$ is a singular matrix, then the value of $9 β$ is _______. [2022]

(3)

$A = (\begin{matrix} β & 0 & 1 \\ 2 & 1 & - 2 \\ 3 & 1 & - 2 \end{matrix}),$ $∴$ $| A |$ $= - 1$

Given that $A^{7} - (β - 1) A^{6} - β A^{5}$ is a singular matrix.

$| A^{5} |$ $| A^{2} - (β - 1) A - β I |$ $= 0$

$\Rightarrow$ $| A^{5} |$ $| (A + I) (A - β I) |$ $= 0$

$∴$ $| A^{5} |$ $| A + I |$ $| A - β I |$ $= 0$ $[∵ | A^{5} | = | A |^{5}]$

As $| A |$ $\neq 0$ , $| A + I |$ $= 0$ $or$ $| A - β I |$ $= 0$

$\Rightarrow$ $| A + I |$ $| \begin{matrix} β + 1 & 0 & 1 \\ 2 & 2 & - 2 \\ 3 & 1 & - 1 \end{matrix} |$ $= 0$

$- 1 = 0$ $(Rejected)$ ${| A + I | \neq 0}$

$∴$ $| A - β I |$ $=$ $| \begin{matrix} 0 & 0 & 1 \\ 2 & 1 - β & - 2 \\ 3 & 1 & - 2 - β \end{matrix} |$ $= 0$

$\Rightarrow 2 - 3 (1 - β) = 0$ $\Rightarrow 2 - 3 + 3 β = 0$

$\Rightarrow β = \frac{1}{3}$

$∴$ $9 β = 3$

Q 5 :

Let $z = \frac{- 1 + \sqrt{3} i}{2},$ where $i = \sqrt{- 1}$ , and $r, s \in {1, 2, 3}$ . Let $P = [\begin{matrix} {(- z)}^{r} & z^{2 s} \\ z^{2 s} & z^{r} \end{matrix}]$ and $I$ be the identity matrix of order 2. Then the total number of ordered pairs $(r, s)$ for which $P^{2} = - I$ is [2016]

(1)

$z = \frac{- 1 + i \sqrt{3}}{2} \Rightarrow z^{3} = 1$ and $1 + z + z^{2} = 0$

$P^{2} = [\begin{matrix} {(- z)}^{r} & z^{2 s} \\ z^{2 s} & z^{r} \end{matrix}] [\begin{matrix} {(- z)}^{r} & z^{2 s} \\ z^{2 s} & z^{r} \end{matrix}]$

$= [\begin{matrix} z^{2 r} + z^{4 s} & z^{2 s} ({(- z)}^{r} + z^{r}) \\ z^{2 s} ({(- z)}^{r} + z^{r}) & z^{4 s} + z^{2 r} \end{matrix}]$

For $P^{2} = - I$ , we should have

$z^{2 r} + z^{4 s} = - 1$ and $z^{2 s} ({(- z)}^{r} + z^{r}) = 0$

$\Rightarrow z^{2 r} + z^{4 s} + 1 = 0$ and ${(- z)}^{r} + z^{r} = 0$

$\Rightarrow r is odd and s = r but not a multiple of 3,$

which is possible when $s = r = 1$

$∴$ $only one pair is there.$

Q 6 :

Let $M$ be a $3 \times 3$ matrix satisfying

$M [\begin{matrix} 0 \\ 1 \\ 0 \end{matrix}] = [\begin{matrix} - 1 \\ 2 \\ 3 \end{matrix}],$ $M [\begin{matrix} 1 \\ - 1 \\ 0 \end{matrix}] = [\begin{matrix} 1 \\ 1 \\ - 1 \end{matrix}],$ and $M [\begin{matrix} 1 \\ 1 \\ 1 \end{matrix}] = [\begin{matrix} 0 \\ 0 \\ 12 \end{matrix}] .$ Then the sum of the diagonal entries of $M$ is [2011]

(9)

Let $M = [\begin{matrix} a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \\ a_{3} & b_{3} & c_{3} \end{matrix}],$ then $[\begin{matrix} a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \\ a_{3} & b_{3} & c_{3} \end{matrix}] [\begin{matrix} 0 \\ 1 \\ 0 \end{matrix}] = [\begin{matrix} - 1 \\ 2 \\ 3 \end{matrix}]$

$\Rightarrow b_{1} = - 1,$ $b_{2} = 2,$ $b_{3} = 3$

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

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

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

and $[\begin{matrix} a_{1} & b_{1} & c_{1} \\ a_{2} & b_{2} & c_{2} \\ a_{3} & b_{3} & c_{3} \end{matrix}] [\begin{matrix} 1 \\ 1 \\ 1 \end{matrix}] = [\begin{matrix} 0 \\ 0 \\ 12 \end{matrix}]$

$\Rightarrow a_{3} + b_{3} + c_{3} = 12$ $\Rightarrow c_{3} = 7$

$∴$ $Sum of diagonal elements = a_{1} + b_{2} + c_{3} = 0 + 2 + 7 = 9$

Q 7 :

Let $ω$ be a complex cube root of unity with $ω \neq 1$ and $P = [p_{i j}]$ be an $n \times n$ matrix with $p_{i j} = ω^{i + j} .$ Then $P^{2} \neq 0,$ when $n =$ [2013]

57
55
58
56

1

2

3

4

Select one or more options

(2, 3, 4)

For $n = 3$ , $P = [\begin{matrix} ω^{2} & ω^{3} & ω^{4} \\ ω^{3} & ω^{4} & ω^{5} \\ ω^{4} & ω^{5} & ω^{6} \end{matrix}]$ and $P^{2} = [\begin{matrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix}]$

It shows $P^{2} = 0$ if $n$ is a multiple of 3.

So, for $P^{2} \neq 0,$ $n$ should not be a multiple of 3, hence $n$ can take values 55, 56 and 58.

Topic Question Set

Q 1 :

If $M = (\begin{matrix} \frac{5}{2} & \frac{3}{2} \\ - \frac{3}{2} & - \frac{1}{2} \end{matrix}),$ then which of the following matrices is equal to $M^{2022} ?$ [2022]

Q 2 :

How many $3 \times 3$ matrices $M$ with entries from ${0, 1, 2}$ are there, for which the sum of the diagonal entries of $M^{T} M$ is 5? [2017]

Q 3 :

Let $P = [\begin{matrix} 1 & 0 & 0 \\ 4 & 1 & 0 \\ 16 & 4 & 1 \end{matrix}]$ and $I$ be the identity matrix of order 3. If $Q = [q_{i j}]$ is a matrix such that $P^{50} - Q = I,$ then $\frac{q_{31} + q_{32}}{q_{21}}$ equals [2016]

Q 4 :

Let $β$ be a real number. Consider the matrix $A = (\begin{matrix} β & 0 & 1 \\ 2 & 1 & - 2 \\ 3 & 1 & - 2 \end{matrix}) .$

If $A^{7} - (β - 1) A^{6} - β A^{5}$ is a singular matrix, then the value of $9 β$ is _______. [2022]

Q 7 :

Let $ω$ be a complex cube root of unity with $ω \neq 1$ and $P = [p_{i j}]$ be an $n \times n$ matrix with $p_{i j} = ω^{i + j} .$ Then $P^{2} \neq 0,$ when $n =$ [2013]

Want Us to Email you About Special Offers & Updates?

Topic Question Set

Q 1 : If M=(5232-32-12), then which of the following matrices is equal to M2022 ? [2022]

Q 2 : How many 3×3 matrices M with entries from {0,1,2} are there, for which the sum of the diagonal entries of MTM is 5? [2017]

Q 3 : Let P=[1004101641] and I be the identity matrix of order 3. If Q=[qij] is a matrix such that P50-Q=I, then q31+q32q21 equals [2016]

Q 4 : Let β be a real number. Consider the matrix A=(β0121-231-2). If A7-(β-1)A6-βA5 is a singular matrix, then the value of 9β is _______. [2022]

Q 5 : Let z=-1+3i2, where i=-1, and r,s∈{1,2,3}. Let P=[(-z)rz2sz2szr] and I be the identity matrix of order 2. Then the total number of ordered pairs (r,s) for which P2=-I is [2016]

Q 6 : Let M be a 3×3 matrix satisfying M[010]=[-123], M[1-10]=[11-1], and M[111]=[0012]. Then the sum of the diagonal entries of M is [2011]

Q 7 : Let ω be a complex cube root of unity with ω≠1 and P=[pij] be an n×n matrix with pij=ω i+j. Then P2≠0, when n= [2013]

Want Us to Email you About Special Offers & Updates?

Q 1 :

If $M = (\begin{matrix} \frac{5}{2} & \frac{3}{2} \\ - \frac{3}{2} & - \frac{1}{2} \end{matrix}),$ then which of the following matrices is equal to $M^{2022} ?$ [2022]

Q 2 :

How many $3 \times 3$ matrices $M$ with entries from ${0, 1, 2}$ are there, for which the sum of the diagonal entries of $M^{T} M$ is 5? [2017]

Q 3 :

Let $P = [\begin{matrix} 1 & 0 & 0 \\ 4 & 1 & 0 \\ 16 & 4 & 1 \end{matrix}]$ and $I$ be the identity matrix of order 3. If $Q = [q_{i j}]$ is a matrix such that $P^{50} - Q = I,$ then $\frac{q_{31} + q_{32}}{q_{21}}$ equals [2016]

Q 4 :

Let $β$ be a real number. Consider the matrix $A = (\begin{matrix} β & 0 & 1 \\ 2 & 1 & - 2 \\ 3 & 1 & - 2 \end{matrix}) .$

If $A^{7} - (β - 1) A^{6} - β A^{5}$ is a singular matrix, then the value of $9 β$ is _______. [2022]

Q 7 :

Let $ω$ be a complex cube root of unity with $ω \neq 1$ and $P = [p_{i j}]$ be an $n \times n$ matrix with $p_{i j} = ω^{i + j} .$ Then $P^{2} \neq 0,$ when $n =$ [2013]