CBSE Class 12 Mathematics 2023: 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. $A^2=A\cdot A=\left[\begin{array}{ccc}1& 2& 3\\ 3& -2& 1\\ 4& 2& 1\end{array}\right]\left[\begin{array}{ccc}1& 2& 3\\ 3& -2& 1\\ 4& 2& 1\end{array}\right]=\left[\begin{array}{ccc}19& 4& 8\\ 1& 12& 8\\ 14& 6& 15\end{array}\right]$

   1 mark

2. $A^3=A^2\cdot A=\left[\begin{array}{ccc}19& 4& 8\\ 1& 12& 8\\ 14& 6& 15\end{array}\right]\left[\begin{array}{ccc}1& 2& 3\\ 3& -2& 1\\ 4& 2& 1\end{array}\right]=\left[\begin{array}{ccc}63& 46& 69\\ 69& -6& 23\\ 92& 46& 63\end{array}\right]$

   1 mark

3. $A^3-23A-40I=\left[\begin{array}{ccc}63& 46& 69\\ 69& -6& 23\\ 92& 46& 63\end{array}\right]-\left[\begin{array}{ccc}23& 46& 69\\ 69& -46& 23\\ 92& 46& 23\end{array}\right]-\left[\begin{array}{ccc}40& 0& 0\\ 0& 40& 0\\ 0& 0& 40\end{array}\right]$

   ½ mark

4. $=\left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]=O$. Hence proved.

   ½ mark

Answer: $A^3-23A-40I=O$ (verified)

Revise the chapter

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