Matrix Algebra PYQ | JEE Main 2025 | Number of Matrices Question

 

❓ Question

Let

$$A=\begin{pmatrix} a & b\\ 0 & d \end{pmatrix}, \qquad a,b,d\in\{-1,0,1\}$$

and

$$(I-A)^3=I-A^3.$$

Find the number of elements in the set of all such matrices $A$.

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✍️ Solution

Using the identity,

$$(I-A)^3=I-3A+3A^2-A^3.$$

Given,

$$(I-A)^3=I-A^3.$$

Hence,

$$I-3A+3A^2-A^3=I-A^3$$
$$3A^2-3A=0$$
$$A^2=A.$$

So, $A$ is idempotent.

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Now,

$$A=\begin{pmatrix} a&b\\ 0&d \end{pmatrix}$$

gives

$$A^2= \begin{pmatrix} a^2 & ab+bd\\ 0 & d^2 \end{pmatrix}.$$

Since $A^2=A$,

$$a=a^2,\qquad d=d^2,\qquad ab+bd=b.$$

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Step 1: Values of $a,d$

Since $a,d\in\{-1,0,1\}$,

$$a=a^2,\ d=d^2$$

implies

$$a,d\in\{0,1\}.$$

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Step 2: Condition on $b$

$$ab+bd=b$$
$$b(a+d-1)=0.$$

Now consider all possibilities:

$a$	$d$	Condition on $b$	No. of choices
0	0	$-b=0\Rightarrow b=0$	1
0	1	$0=0$	3
1	0	$0=0$	3
1	1	$b=0$	1

Total matrices

$$1+3+3+1=8.$$

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✅ Final Answer

$$\boxed{8}$$