Math Problem Statement
Find the determinant of the given 3x3 matrix: [-3 4 -7; -4 -2 7; -6 5 9].
Solution
The image shows a 3x3 matrix, and you are tasked with finding its determinant. The matrix, , is:
-3 & 4 & -7 \\ -4 & -2 & 7 \\ -6 & 5 & 9 \end{bmatrix}$$ To find the determinant of a 3x3 matrix $$A = \begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix}$$, we use the following formula: $$\text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg)$$ For the given matrix, the elements are: $$a = -3, \, b = 4, \, c = -7, \, d = -4, \, e = -2, \, f = 7, \, g = -6, \, h = 5, \, i = 9$$ Substituting these values into the determinant formula: $$\text{det}(\Delta) = -3((-2)(9) - (7)(5)) - 4((-4)(9) - (7)(-6)) + (-7)((-4)(5) - (-2)(-6))$$ I'll now calculate the result.The determinant of the given matrix $$\Delta$$ is: $$\text{det}(\Delta) = 359$$ Would you like further details on the steps or calculations involved? Here are some related questions to explore: 1. What is the geometric significance of the determinant of a matrix? 2. How would the determinant change if one row of the matrix is multiplied by a scalar? 3. What does it mean when the determinant of a matrix is zero? 4. How do cofactor expansion and row reduction methods compare for finding determinants? 5. Can determinants be used to solve systems of linear equations? **Tip:** For larger matrices, finding determinants via row reduction or Laplace expansion is more efficient than directly applying the formula.Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Linear Algebra
Determinants
Matrices
Formulas
Determinant of a 3x3 matrix: det(A) = a(ei - fh) - b(di - fg) + c(dh - eg)
Theorems
Determinant Theorem
Cofactor Expansion
Suitable Grade Level
Grades 11-12, College Level