Calculate the Determinant of a 3x3 Matrix | Example with Detailed Solution

Math Problem Statement

ypočítajte determinant

−5 2 0

6 −1 4

1 3 3

To calculate the determinant of the 3x3 matrix

-5 & 2 & 0 \\ 6 & -1 & 4 \\ 1 & 3 & 3 \\ \end{vmatrix}$$ we use the formula for a 3x3 matrix determinant: $$\text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg)$$ where $$a, b, c, d, e, f, g, h,$$ and $$i$$ are elements of the matrix: $$\begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \\ \end{bmatrix}$$ In this matrix: - $$a = -5$$, $$b = 2$$, $$c = 0$$ - $$d = 6$$, $$e = -1$$, $$f = 4$$ - $$g = 1$$, $$h = 3$$, $$i = 3$$ Substituting into the formula, we get: $$\text{det}(A) = -5((-1)(3) - (4)(3)) - 2((6)(3) - (4)(1)) + 0((6)(3) - (-1)(1))$$ Calculating each term: 1. $$(-1)(3) - (4)(3) = -3 - 12 = -15$$ 2. $$(6)(3) - (4)(1) = 18 - 4 = 14$$ Thus, $$\text{det}(A) = -5(-15) - 2(14) + 0 = 75 - 28 + 0 = 47$$ The determinant of the matrix is: $$\text{det}(A) = 47$$ Would you like further details or have any questions? --- Here are some related questions: 1. What is the general process for finding determinants of larger matrices? 2. How does the determinant change if a row or column is multiplied by a scalar? 3. How are determinants used in solving systems of equations? 4. What does a zero determinant indicate about a matrix? 5. Can we calculate determinants for non-square matrices? **Tip:** Always double-check signs carefully when expanding determinants, especially with negative values.

Drop file here or Click Here to upload

Linear Algebra
Determinants
Matrix Operations

3x3 determinant formula: det(A) = a(ei - fh) - b(di - fg) + c(dh - eg)

Determinant properties for 3x3 matrices

Grades 10-12

Calculate the Determinant of a 3x3 Matrix: Step-by-Step Solution

Calculate Determinant of a 3x3 Matrix: Detailed Step-by-Step Solution

Evaluate the Determinants of 3x3 Matrices - Step-by-Step

Calculating the Determinant of a 3x3 Matrix: 1 -1 3, 2 2 1, 2 1 1

Find the Determinant of 3x3 Matrices: Step-by-Step Solution