Let and be real numbers. Consider a 3 3 matrix A such that: A^2 = 3A + I, A^4 = 21A + I. Then:

Given: A^2 = 3A + I and A^4 = 21A + I Find: Which statement is correct. From A^2 = 3A + I multiply by A to get A^3 = A A^2 = A(3A + I) = 3A^2 + A Now substitute A^2 = 3A + I so A^3 = 3(3A + I) + A = 9A + 3 I + A Therefore, A^3 = (9 +)A + 3 I Now compute A^4 using the expression for A^3: A^4 = A A^3 = A ((9 +)A + 3 I) = (9 +)A^2 + 3 A Substitute again from A^2 = 3A + I Then A^4 = (9 +)(3A + I) + 3 A A^4 = (27 + 3)A + (9 +)I + 3 A A^4 = (27 + 6)A + (9 +)I Compare this with the given equation A^4 = 21A + I So, 27 + 6 = 21 6 = -6 = -1 and = (9 +) = (-1)(8) = -8 Therefore, the correct option is D and = -8.

Let and be real numbers. Consider a 3 3 matrix A such that: A^2 =… — Solution | JEE Mathematics

MCQMediumJEE 2023Algebra of Matrices

Let $\alpha$ and $\beta$ be real numbers. Consider a $3 \times 3$ matrix $A$ such that:

A^2 = 3A + \alpha I,

A^4 = 21A + \beta I.

Then:

A
$\alpha = 1$
B
$\alpha = 4$
C
$\beta = 8$
D
$\beta = -8$

Answer

Correct answer:D

Step-by-step solution

Standard Method

Given:

A^2 = 3A + \alpha I

and

A^4 = 21A + \beta I

Find: Which statement is correct.

From

A^2 = 3A + \alpha I

multiply by $A$ to get

A^3 = A \cdot A^2 = A(3A + \alpha I) = 3A^2 + \alpha A

Now substitute

A^2 = 3A + \alpha I

A^3 = 3(3A + \alpha I) + \alpha A = 9A + 3\alpha I + \alpha A

Therefore,

A^3 = (9 + \alpha)A + 3\alpha I

Coefficient Comparison

Now compute $A^4$ using the expression for $A^3$ :

A^4 = A \cdot A^3 = A\big((9 + \alpha)A + 3\alpha I\big) = (9 + \alpha)A^2 + 3\alpha A

Substitute again from

A^2 = 3A + \alpha I

Then

A^4 = (9 + \alpha)(3A + \alpha I) + 3\alpha A

A^4 = (27 + 3\alpha)A + \alpha(9 + \alpha)I + 3\alpha A

A^4 = (27 + 6\alpha)A + \alpha(9 + \alpha)I

Compare this with the given equation

A^4 = 21A + \beta I

So,

27 + 6\alpha = 21 \Rightarrow 6\alpha = -6 \Rightarrow \alpha = -1

and

\beta = \alpha(9 + \alpha) = (-1)(8) = -8

Therefore, the correct option is D and $\beta = -8$ .

Common mistakes

Students may compare only the coefficient of $A$ and forget to compare the coefficient of $I$ . Since both matrices are expressed as linear combinations of $A$ and $I$ , both coefficients must match.
A common mistake is expanding $A^3$ incorrectly as $9A + 3\alpha + \alpha A$ , treating $I$ like a scalar term. The identity matrix must remain attached as $3\alpha I$ .
Some students square the first equation carelessly and introduce unnecessary higher powers without reducing them back using $A^2 = 3A + \alpha I$ . The correct approach is to reduce every higher power to a combination of $A$ and $I$ .

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Answer

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