Let α and β be real numbers. Consider a 3 × 3 matrix A such that A² = 3A + αI. If A⁴ = 21A + βI, the

Q: Why can't we just find the eigenvalues of $A$?

You absolutely could! If $\lambda$ is an eigenvalue of $A$, then $\lambda^2 = 3\lambda + \alpha$ and $\lambda^4 = 21\lambda + \beta$. This would give you a system of equations. However, for JEE Advanced, mastering the polynomial identity method is often faster and less prone to calculation errors with roots. It builds a deeper intuition for matrix algebra.

Q: Wait, why do we substitute $A^2$ back into the equation?

Great question! When we square $A^2$, we get an $A^2$ term in our expansion. But the target equation $A^4 = 21A + \beta I$ only contains $A$ and $I$. By substituting $A^2 = 3A + \alpha I$ back in, we reduce the degree of the polynomial, effectively 'collapsing' the $A^2$ term into linear terms of $A$ and $I$.

Q: Is it always true that $A$ and $I$ commute?

Yes, absolutely! The identity matrix $I$ acts like the number $1$ in scalar multiplication. For any square matrix $A$, $AI = IA = A$. This property is the bedrock of matrix polynomial algebra, allowing us to treat $A$ and $I$ like variables in a standard algebraic expression.

Question

Let $\alpha$ and $\beta$ be real numbers. Consider a $3 	imes 3$ matrix $A$ such that $A^2 = 3A + \alpha I$. If $A^4 = 21A + \beta I$, then

Options:

(A) $\alpha = 1$

(B) $\alpha = 4$

(C) $\beta = 8$

(D) $\beta = -8$

Sigma Percentile · Accepted Answer

### Analyzing the Setup

We are given a $3 	imes 3$ matrix $A$ that satisfies two specific matrix polynomial equations:
1. $A^2 = 3A + \alpha I$
2. $A^4 = 21A + \beta I$

Our objective is to determine the values of the constants $\alpha$ and $\beta$ by treating the matrix $A$ as an algebraic variable.

### The Bridge

The relationship between $A^2$ and $A^4$ is defined by the identity $A^4 = (A^2)^2$. To bridge the two given equations, we square the first equation:

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

This substitution allows us to express $A^4$ in terms of $A$ and $I$ using binomial expansion.

### The Expansion

Since the matrix $A$ and the identity matrix $I$ commute, we expand the right-hand side using the standard algebraic identity $(x + y)^2 = x^2 + 2xy + y^2$:

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

Simplifying this expression, and noting that $AI = A$ and $I^2 = I$, we obtain:

$$A^4 = 9A^2 + 6\alpha A + \alpha^2 I$$

### The Reduction

To align this result with the target equation $A^4 = 21A + \beta I$, we must eliminate the $A^2$ term. We substitute the original definition $A^2 = 3A + \alpha I$ into our expanded expression:

$$A^4 = 9(3A + \alpha I) + 6\alpha A + \alpha^2 I$$

Distributing the constant $9$ across the terms, we get:

$$A^4 = 27A + 9\alpha I + 6\alpha A + \alpha^2 I$$

### Final Calculation

Grouping the terms involving $A$ and the terms involving $I$, we arrive at:

$$A^4 = (27 + 6\alpha)A + (9\alpha + \alpha^2)I$$

By comparing this to the given equation $A^4 = 21A + \beta I$, we equate the coefficients:

1. $27 + 6\alpha = 21 \Rightarrow 6\alpha = -6 \Rightarrow \alpha = -1$
2. $\beta = 9\alpha + \alpha^2$

Substituting $\alpha = -1$ into the second equation:

$$\beta = 9(-1) + (-1)^2 = -9 + 1 = -8$$

The final values are **$\alpha = -1$** and **$\beta = -8$**.

Animated Solution for Mathematics - Matrices and Determinants: Let $α$ and $β$ be real numbers. Consider a $3 \times 3$ matrix $A$ such that $A^{2} = 3 A + α I$ . If $A^{4} = 21 A + β I$ , then

Select Answer:

Visualized Solution

Analyze the Given Matrix Equations

Square the First Equation

Expand the Algebraic Square

Simplify the Expansion

Substitute $A^{2}$ Back

Distribute and Simplify

Group Terms of $A$ and $I$

Compare with the Given $A^{4}$ Equation

Solve for $α$

Solve for $β$

Final Conclusion

The Sigma Insight: Algebraic Operations on Matrices

Analyzing the Setup

The Bridge

The Expansion

The Reduction

Final Calculation

Similar Questions

Suppose A is any $3 \times 3$ non-singular matrix and $(A - 3 I) (A - 5 I) = 0$ , where $I = I_{3}$ and $O = O_{3}$ . If $α A + β A^{- 1} = 4 I$ , then $α + β$ is equal to :-

Let $A$ be a $3 \times 3$ real matrix such that $A^{2} (A - 2 I) - 4 (A - I) = O$ , where $I$ and $O$ are the identity and null matrices, respectively. If $A^{5} = α A^{2} + β A + γ I$ , where $α, β$ and $γ$ are real constants, then $α + β + γ$ is equal to:

If the matrix $A = 103020 00 - 1$ satisfies the equation $A^{20} + α A^{19} + β A = 100040001$ for some real numbers $α$ and $β$ , then $β - α$ is equal to

If for the matrix, $A = [1 α - α β]$ , $A A^{T} = I_{2}$ , then the value of $α^{4} + β^{4}$ is:

Let $A = 210 a 35 01 b$ . If $A^{3} = 4 A^{2} - A - 21 I$ , where $I$ is the identity matrix of order $3 \times 3$ , then $2 a + 3 b$ is equal to:

Let $A$ be a $2 \times 2$ matrix with real entries such that $A^{T} = α A + I$ , where $α \in R - {- 1, 1}$ . If $det (A^{2} - A) = 4$ , the sum of all possible values of $α$ is equal to

Let $B = [1135]$ and $A$ be a $2 \times 2$ matrix such that $A B^{- 1} = A^{- 1}$ . If $B C B^{- 1} = A$ and $C^{4} + α C^{2} + β I = O$ , then $2 β - α$ is equal to

Let $P$ be a square matrix such that $P^{2} = I - P$ . For $α, β, γ, δ \in N$ , if $P^{α} + P^{β} = γ I - 29 P$ and $P^{α} - P^{β} = δ I - 13 P$ , then $α + β + γ - δ$ is equal to

Let $A = [a_{ij}]$ be $3 \times 3$ matrix such that $A 010 = 001$ , $A 413 = 010$ and $A 212 = 100$ , then $a_{23}$ equals:

If $A = [a b b a]$ and $A^{2} = [α β β α]$ , then

Animated Solution for Mathematics - Matrices and Determinants: .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let α and β be real numbers. Consider a 3×3 matrix A such that A2=3A+αI. If A4=21A+βI, then

Select Answer:

Visualized Solution

Analyze the Given Matrix Equations

Square the First Equation

Expand the Algebraic Square

Simplify the Expansion

Substitute .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } A2 Back

Distribute and Simplify

Group Terms of .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } A and .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } I

Compare with the Given .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } A4 Equation

Solve for .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } α

Solve for .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } β