Math Problem Statement

Let A = [[2, 1, 91], [a, 1, 2], [0, 2, 5]]. Find the determinant of matrix A.

Solution

The given matrix AA is a 3×33 \times 3 matrix:

2 & 1 & 91 \\ a & 1 & 2 \\ 0 & 2 & 5 \end{pmatrix}$$ We are asked to compute the determinant of matrix $$A$$. The formula for the determinant of a $$3 \times 3$$ matrix is: $$\text{det}(A) = A_{11} \cdot \text{det}\begin{pmatrix} A_{22} & A_{23} \\ A_{32} & A_{33} \end{pmatrix} - A_{12} \cdot \text{det}\begin{pmatrix} A_{21} & A_{23} \\ A_{31} & A_{33} \end{pmatrix} + A_{13} \cdot \text{det}\begin{pmatrix} A_{21} & A_{22} \\ A_{31} & A_{32} \end{pmatrix}$$ Here, $$A_{ij}$$ represents the element in the $$i$$-th row and $$j$$-th column of matrix $$A$$. Let's compute the determinant using cofactor expansion: ### Step 1: Expansion along the first row $$\text{det}(A) = 2 \cdot \text{det} \begin{pmatrix} 1 & 2 \\ 2 & 5 \end{pmatrix} - 1 \cdot \text{det} \begin{pmatrix} a & 2 \\ 0 & 5 \end{pmatrix} + 91 \cdot \text{det} \begin{pmatrix} a & 1 \\ 0 & 2 \end{pmatrix}$$ ### Step 2: Calculate the $$2 \times 2$$ determinants $$\text{det} \begin{pmatrix} 1 & 2 \\ 2 & 5 \end{pmatrix} = (1)(5) - (2)(2) = 5 - 4 = 1$$ $$\text{det} \begin{pmatrix} a & 2 \\ 0 & 5 \end{pmatrix} = (a)(5) - (2)(0) = 5a$$ $$\text{det} \begin{pmatrix} a & 1 \\ 0 & 2 \end{pmatrix} = (a)(2) - (1)(0) = 2a$$ ### Step 3: Substitute back into the formula $$\text{det}(A) = 2 \cdot 1 - 1 \cdot 5a + 91 \cdot 2a$$ $$\text{det}(A) = 2 - 5a + 182a$$ $$\text{det}(A) = 2 + 177a$$ Thus, the determinant of matrix $$A$$ is: $$\boxed{2 + 177a}$$ ### Do you want details on any steps or have questions? Here are some related questions: 1. How do you compute the cofactor expansion for larger matrices? 2. How does the determinant affect the invertibility of a matrix? 3. Can a matrix with a variable element in the determinant have multiple values for specific cases? 4. How do determinants relate to eigenvalues of a matrix? 5. How does row or column swapping affect the determinant? **Tip:** The determinant of a matrix can often be simplified by choosing a row or column with many zeros for cofactor expansion, reducing calculation complexity.

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Determinants

Formulas

Determinant of a 3x3 matrix: det(A) = A11 * det([[A22, A23], [A32, A33]]) - A12 * det([[A21, A23], [A31, A33]]) + A13 * det([[A21, A22], [A31, A32]])

Theorems

Cofactor Expansion Theorem

Suitable Grade Level

Grades 10-12