Answer
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Hint:For solving this type of question, we consider the given data ${P^3} = {Q^3}$, ${P^2}Q = P{Q^2}$ as equation 1 and 2 respectively. Then we subtract both the equations with each other and we get the value of ${P^2} + {Q^2}$ After this we find the determinant of ${P^2} + {Q^2}$.
Complete step-by-step answer:
According to the question it is given that P and Q are the matrices of order $3 \times 3$. And $P \ne Q$
It is also given that
$ \Rightarrow {P^3} = {Q^3} - - - - - (1)$
and
$ \Rightarrow {P^2}Q = P{Q^2} - - - - - (2)$
We find the value of $\left| {{P^2} + {Q^2}} \right|$ for this we subtract equation (2) from (1), we get
$ \Rightarrow {P^3} - {P^2}Q = {Q^3} - {Q^2}P$
Now taking ${P^2}$ common from L.H.S and -${Q^2}$ common from R.H.S, we get
$ \Rightarrow {P^2}(P - Q) = - {Q^2}(P - Q)$
Now , we take the R.H.S terms to the L.H.S side we get,
$ \Rightarrow {P^2}(P - Q) + {Q^2}(P - Q) = 0$
Now we take $(P - Q)$ common, we get
$ \Rightarrow (P - Q)({P^2} + {Q^2}) = 0$
Now it is given that $P \ne Q$
So, $(P - Q) \ne 0$
Therefore from equation we get
$\left| {{P^2} + {Q^2}} \right| = 0$
The value of determinant of $({P^2} + {Q^2})$ is 0
So, the correct answer is “Option A”.
Additional Information:Matrix: It is an array of many numbers. A square matrix is a matrix which has the number of rows and columns equal.
Determinant of a matrix: It is a special number that can be calculated from a square matrix. The determinant of a matrix P is denoted by det(P) or $\left| P \right|$.
Note:For solving these kinds of questions we have to always remember the conditions given to us and compute these two equations and the matrices should be of the same order.
Complete step-by-step answer:
According to the question it is given that P and Q are the matrices of order $3 \times 3$. And $P \ne Q$
It is also given that
$ \Rightarrow {P^3} = {Q^3} - - - - - (1)$
and
$ \Rightarrow {P^2}Q = P{Q^2} - - - - - (2)$
We find the value of $\left| {{P^2} + {Q^2}} \right|$ for this we subtract equation (2) from (1), we get
$ \Rightarrow {P^3} - {P^2}Q = {Q^3} - {Q^2}P$
Now taking ${P^2}$ common from L.H.S and -${Q^2}$ common from R.H.S, we get
$ \Rightarrow {P^2}(P - Q) = - {Q^2}(P - Q)$
Now , we take the R.H.S terms to the L.H.S side we get,
$ \Rightarrow {P^2}(P - Q) + {Q^2}(P - Q) = 0$
Now we take $(P - Q)$ common, we get
$ \Rightarrow (P - Q)({P^2} + {Q^2}) = 0$
Now it is given that $P \ne Q$
So, $(P - Q) \ne 0$
Therefore from equation we get
$\left| {{P^2} + {Q^2}} \right| = 0$
The value of determinant of $({P^2} + {Q^2})$ is 0
So, the correct answer is “Option A”.
Additional Information:Matrix: It is an array of many numbers. A square matrix is a matrix which has the number of rows and columns equal.
Determinant of a matrix: It is a special number that can be calculated from a square matrix. The determinant of a matrix P is denoted by det(P) or $\left| P \right|$.
Note:For solving these kinds of questions we have to always remember the conditions given to us and compute these two equations and the matrices should be of the same order.
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