Calculate the Determinants of 2x2 and 3x3 Matrices

Math Problem Statement

Evaluate the determinant of the given matrices: 5) A 2x2 matrix [3, 2; 4, -3], 6) A 3x3 matrix [1, 5, -4; 5, 2, -3; -3, -2, 3]

Let's evaluate the determinant for each of the matrices provided in the image.

3 & 2 \\ 4 & -3 \end{bmatrix}$$ The determinant is calculated as follows: $$\text{det}(A) = (3 \times -3) - (2 \times 4) = -9 - 8 = -17$$ 2. For the 3x3 matrix: $$\begin{bmatrix} 1 & 5 & -4 \\ 5 & 2 & -3 \\ -3 & -2 & 3 \end{bmatrix}$$ The determinant for a 3x3 matrix, $$A$$, can be calculated using the rule of Sarrus or cofactor expansion. We will use the rule of Sarrus here: $$\text{det}(A) = (1 \times 2 \times 3) + (5 \times -3 \times -3) + (-4 \times 5 \times -2) - (-4 \times 2 \times -3) - (1 \times -3 \times -2) - (5 \times 5 \times 3)$$ Calculating this gives: $$\text{det}(A) = (6) + (45) + (40) - (24) - (6) - (75)$$ $$= 91 - 105 = -14$$ So the determinants of the matrices are: - For the 2x2 matrix: $$-17$$ - For the 3x3 matrix: $$-14$$ Would you like more details on any of these steps, or do you have any other questions? Here are some additional related questions you might consider: 1. How does changing an element in a matrix affect its determinant? 2. What is the geometric interpretation of a determinant? 3. How would adding a multiple of one row to another row affect the determinant? 4. Can you explain how the determinant relates to the invertibility of a matrix? 5. What happens to the determinant when you transpose a matrix? **Tip:** Remember, the determinant of a matrix is zero if and only if the matrix is singular, meaning it does not have an inverse.

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Linear Algebra
Matrix Determinant

Determinant of a 2x2 matrix: det(A) = ad - bc
Determinant of a 3x3 matrix using Sarrus' Rule: det(A) = aei + bfg + cdh - afh - bdi - ceg

Sarrus' Rule
Determinant of a matrix

Grades 10-12 (High School)

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How to Calculate the Determinant of a 2x2 Matrix [6, 8; 3, 7]