CBSE Class 12 Mathematics sample paper 2026-27: question 26

Hints

Open one hint at a time and try again before the next.

Step-by-step solution

Each step shows the marks it earns, as an examiner would award them. Checked by an independent re-solve from the question paper.

1. $P$ is $2\times 3$ and $Q$ is $3\times 2$, so both products are defined. $PQ=\left[\begin{array}{ccc}2& 0& 0\\ 0& 1& 2\end{array}\right]\left[\begin{array}{cc}1& 0\\ 0& 2\\ 1& 0\end{array}\right]=\left[\begin{array}{cc}2& 0\\ 2& 2\end{array}\right]$

   1 mark

2. $QP=\left[\begin{array}{cc}1& 0\\ 0& 2\\ 1& 0\end{array}\right]\left[\begin{array}{ccc}2& 0& 0\\ 0& 1& 2\end{array}\right]=\left[\begin{array}{ccc}2& 0& 0\\ 0& 2& 4\\ 2& 0& 0\end{array}\right]$

   1 mark

3. $|PQ|=4-0=4\ne 0$, so $PQ$ is non-singular and its inverse exists. $(PQ{)}^{-1}=\frac{1}{|PQ|}\text{adj}(PQ)=\frac{1}{4}\left[\begin{array}{cc}2& 0\\ -2& 2\end{array}\right]=\left[\begin{array}{cc}\frac{1}{2}& 0\\ -\frac{1}{2}& \frac{1}{2}\end{array}\right]$

   ½ mark

4. $|QP|=0$ because rows 1 and 3 are identical, so $QP$ is singular and its inverse does not exist.

   ½ mark

Answer: $PQ=\left[\begin{array}{cc}2& 0\\ 2& 2\end{array}\right]$, $(PQ{)}^{-1}=\left[\begin{array}{cc}\frac{1}{2}& 0\\ -\frac{1}{2}& \frac{1}{2}\end{array}\right]$; $QP=\left[\begin{array}{ccc}2& 0& 0\\ 0& 2& 4\\ 2& 0& 0\end{array}\right]$ is singular, so its inverse does not exist

Revise the chapter

• Chapter: Matrices (notes, formulas, where marks are lost and every past question)