If P and Q are two invertible matrices of the | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(A) We know that for any invertible matrix $A$,$\operatorname{Adj}(A) = |A| A^{-1}$.Given that $P$ and $Q$ are invertible matrices,their product $QP$

Vedclass · Accepted Answer

$\operatorname{Adj}(P) \operatorname{Adj}(Q)$

Vedclass · Answer

(A) We know that for any invertible matrix $A$,$\operatorname{Adj}(A) = |A| A^{-1}$.Given that $P$ and $Q$ are invertible matrices,their product $QP$ is also an invertible matrix.Therefore,$\operatorname{Adj}(QP) = |QP| (QP)^{-1}$.Using the properties of determinants,$|QP| = |Q||P| = |P||Q|$.Using the property of inverse matrices,$(QP)^{-1} = P^{-1} Q^{-1}$.Thus,$\operatorname{Adj}(QP) = |P||Q| P^{-1} Q^{-1}$.However,note that $\operatorname{Adj}(Q) = |Q| Q^{-1}$ and $\operatorname{Adj}(P) = |P| P^{-1}$.Multiplying these,we get $\operatorname{Adj}(Q) \operatorname{Adj}(P) = (|Q| Q^{-1}) (|P| P^{-1}) = |Q||P| Q^{-1} P^{-1}$.Since $\operatorname{Adj}(QP) = |QP| (QP)^{-1} = |Q||P| P^{-1} Q^{-1}$,and generally $P^{-1} Q^{-1} 
eq Q^{-1} P^{-1}$,the standard identity is $\operatorname{Adj}(QP) = \operatorname{Adj}(P) \operatorname{Adj}(Q)$.

If $P$ and $Q$ are two invertible matrices of the same order,then $\operatorname{Adj}(QP)$ is equal to

Similar Questions

If $A$ is a square matrix of order $3$ and $|A|=5$,then $|A \text{ adj. } A|$ is

The inverse of the matrix $\left[\begin{array}{ccc}7 & -3 & -3 \\ -1 & 1 & 0 \\ -1 & 0 & 1\end{array}\right]$ is

Find the inverse of the matrix,if it exists: $\left[\begin{array}{ll}2 & -6 \\ 1 & -2\end{array}\right]$

If $A = \begin{bmatrix} 2 & 3 \\ 1 & 2 \end{bmatrix}$ and $B = \begin{bmatrix} 1 & 0 \\ 3 & 1 \end{bmatrix}$,then $(AB)^{-1} =$

If $A = \begin{bmatrix} a & c \\ d & b \end{bmatrix}$,then $A^{-1} = $

Vedclass Test Series

Exam Paper Generator

Online Exam Module