CBSE Board Examination 2023 · Section A · 1 mark

Q1: $A^{-1} = xA + yI$ when $A^2 - 3A + I = O$

Board 2023Ch 4 · Determinants Ch 3 · Matrices

The question

If for a square matrix A, $A^2 - 3A + I = O$ and $A^{-1} = xA + yI$ ⁠, then the value of $x + y$ is :

(a) $-2$

(b) $2$

(d) $-3$

Exam answer

Multiplying $A^2 - 3A + I = O$ by $A^{-1}$ ⁠:

A^{-1}A^2 - 3A^{-1}A + A^{-1}I = O

Why?

We want $A^{-1}$ on its own, and multiplying by it is the way to make it appear. $A^{-1}$ exists: the question gives it, and the equation itself says $A(3I - A) = I$ ⁠.

Multiply every term on the same side (the left) by $A^{-1}$ ⁠, then simplify each term using $A^{-1}A = I$ ⁠:

\begin{aligned} A^{-1}A^2 &= (A^{-1}A)A \\ &= IA = A \end{aligned}

Also $3A^{-1}A = 3I$ ⁠, $A^{-1}I = A^{-1}$ and $A^{-1}O = O$ ⁠.

\begin{aligned} &A - 3I + A^{-1} = O \\ \Rightarrow\ &A^{-1} = -A + 3I \end{aligned}

Why?

Move $A$ and $-3I$ to the other side, changing their signs.

\begin{aligned} A - 3I + A^{-1} &= O \\ A^{-1} &= O - A + 3I \\ A^{-1} &= -A + 3I \end{aligned}

Watch out: write $3I$ ⁠, not $3$ ⁠. A matrix cannot be added to a number.

Comparing with $A^{-1} = xA + yI$ ⁠: $x = -1$ ⁠, $y = 3$ and $x + y = 2$ ⁠.

Why?

Match the two forms term by term: the number in front of $A$ is $x$ ⁠, and the number in front of $I$ is $y$ ⁠.

\begin{aligned} -A + 3I &= (-1)A + 3I \\ &= xA + yI \end{aligned}

So $x = -1$ and $y = 3$ ⁠. Add them:

\begin{aligned} x + y &= (-1) + 3 \\ &= 2 \end{aligned}

Answer: (b) $2$

Where this comes from

Chapter 4, Determinants: Inverse of a matrix from a matrix equation.

Question type: A matrix equation, then "hence find $A^{-1}$ "

How to spot it: click to see

Type 7. A matrix equation, then "hence find $A^{-1}$ "

Spot it: "show that $A^2 - 5A + 7I = O$ " (or a similar equation), then "hence find $A^{-1}$ ".

Here: The equation $A^2 - 3A + I = O$ is given: multiply it by $A^{-1}$ to get $A - 3I + A^{-1} = O$ ⁠, so $A^{-1} = -A + 3I$ ⁠, with $x = -1$ and $y = 3$ ⁠.

Quick lesson · Worked NCERT solutions · Interactive lesson

Also uses Chapter 3, Matrices.

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Determinants in past papers: question 10 of 34 · see them all

Q1: A−1=xA+yIA^{-1} = xA + yIA−1=xA+yI when A2−3A+I=OA^2 - 3A + I = OA2−3A+I=O

Q1: $A^{-1} = xA + yI$ when $A^2 - 3A + I = O$