If $A = \begin{bmatrix} 2 & 3 \\ 3 & 5 \end{bmatrix}$, then the determinant of the matrix $(A^{2025} – 3A^{2024} + A^{2023})$ is
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Solution Steps
1
Characteristic Equation
For matrix $A = \begin{bmatrix} 2 & 3 \\ 3 & 5 \end{bmatrix}$, calculate Trace and Determinant:
$\text{Trace}(A) = 2 + 5 = 7$
$|A| = (2)(5) – (3)(3) = 10 – 9 = 1$
The characteristic equation is $\lambda^2 – 7\lambda + 1 = 0$.
2
Apply Cayley-Hamilton Theorem
A matrix satisfies its own characteristic equation:
$A^2 – 7A + I = O \implies A^2 = 7A – I$
This will be used to simplify the high power expression.
3
Factor the Matrix Expression
Expression: $E = A^{2025} – 3A^{2024} + A^{2023}$
Factor out $A^{2023}$:
$E = A^{2023}(A^2 – 3A + I)$
4
Substitute $A^2$ into the expression
Substitute $A^2 = 7A – I$ from Step 2:
$E = A^{2023}(7A – I – 3A + I)$
$E = A^{2023}(4A)$
$E = 4A^{2024}$
5
Calculate Determinant of $E$
We need $|E| = |4A^{2024}|$.
Using property $|kA| = k^n|A|$ for $n \times n$ matrix (here $n=2$):
$|4A^{2024}| = 4^2 |A^{2024}|$
6
Final Evaluation
Since $|A| = 1$, then $|A^{2024}| = |A|^{2024} = 1^{2024} = 1$.
$|E| = 16 \times 1 = 16$.
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Key Insight
Cayley-Hamilton Theorem is the most powerful tool for simplifying polynomial expressions of matrices with high powers.
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Matrix Theory
1. Cayley-Hamilton Theorem
The Cayley-Hamilton theorem states that every square matrix $A$ satisfies its characteristic equation $p(\lambda) = \det(A – \lambda I) = 0$. For a $2 \times 2$ matrix, this equation is $\lambda^2 – \text{Tr}(A)\lambda + \det(A) = 0$. This allows us to express higher powers of $A$ as a linear combination of $A$ and $I$, which is crucial for simplifying complex expressions like the one in this problem.
2. Properties of Determinants
Determinants follow several multiplicative properties. For any $n \times n$ matrix $A$ and scalar $k$, the determinant of the scaled matrix is $|kA| = k^n|A|$. Additionally, for any two square matrices $A$ and $B$, $|AB| = |A||B|$. This leads to the result that $|A^n| = |A|^n$. In this question, since $|A|=1$, raising the matrix to the power of 2024 did not change the numerical value of its determinant.
3. Trace and Matrix Algebra
The trace of a matrix, denoted $\text{Tr}(A)$, is the sum of the elements on the main diagonal. For a $2 \times 2$ matrix $A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$, $\text{Tr}(A) = a + d$. The trace and the determinant together define the characteristic polynomial of a $2 \times 2$ matrix completely. Recognizing these properties quickly allows for rapid calculation of matrix identities without performing full matrix multiplication.
4. Reducing Matrix Polynomials
When faced with a high-power matrix polynomial $P(A)$, one can use polynomial division. If $p(A) = O$ is the characteristic equation, then $P(A) = q(A)p(A) + r(A)$. Since $p(A)$ is the zero matrix, $P(A)$ simplifies directly to the remainder $r(A)$, which is of lower degree than the characteristic equation. This “remainder theorem” for matrices simplifies computations significantly.
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FAQs
Q
Why did $|4A^{2024}|$ become $4^2$?
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Because A is a 2×2 matrix. The property is $|kA| = k^n|A|$, where n is the order. Since n=2, we get 4² = 16.
Q
What if $|A|$ was -1?
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Then $|A^{2024}| = (-1)^{2024} = 1$. If the power was odd, it would be -1.
Q
Can I use this for 3×3 matrices?
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Yes, but the characteristic equation will be a cubic: $\lambda^3 – \text{Tr}(A)\lambda^2 + … – |A| = 0$.
Q
How to calculate |A| quickly?
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Multiply the main diagonal elements and subtract the product of the off-diagonal elements: $ad – bc$.
Q
Is A symmetric here?
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Yes, because the off-diagonal elements are both 3 ($A_{12} = A_{21}$).
Q
What is ‘I’ in the equations?
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I is the Identity Matrix, $\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}$, which acts like the number 1 in matrix algebra.
Q
Why factor out $A^{2023}$?
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It is the lowest power in the expression, making it the easiest term to factor out to isolate the quadratic part.
Q
Can this be solved by eigenvalues?
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Yes, if λ is an eigenvalue of A, then λ²⁰²³(λ² – 3λ + 1) is an eigenvalue of the expression matrix.
Q
What does ‘O’ mean in Step 2?
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O represents the Null Matrix or Zero Matrix, where all elements are zero.
Q
Is $|A^{2025}|$ always $|A|^{2025}$?
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Yes, this is a fundamental property of determinants for any square matrix and integer power.
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