Math Problem Statement

Given matrix A = [[6, 2, 3], [0, 5, 0], [4, 8, 1]], calculate det(A).

Solution

The given matrix AA is:

6 & 2 & 3 \\ 0 & 5 & 0 \\ 4 & 8 & 1 \end{pmatrix}$$ To calculate $$\det(A)$$, we use the determinant formula for a 3x3 matrix: $$\det(A) = a(ei − fh) − b(di − fg) + c(dh − eg)$$ Here: $$A = \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix}$$ Substituting the values: $$a = 6, b = 2, c = 3, d = 0, e = 5, f = 0, g = 4, h = 8, i = 1$$ ### Step-by-step Calculation 1. Compute $$ei - fh$$: $$ei - fh = (5 \cdot 1) - (0 \cdot 8) = 5$$ 2. Compute $$di - fg$$: $$di - fg = (0 \cdot 1) - (0 \cdot 4) = 0$$ 3. Compute $$dh - eg$$: $$dh - eg = (0 \cdot 8) - (5 \cdot 4) = -20$$ 4. Substitute into the determinant formula: $$\det(A) = 6(5) - 2(0) + 3(-20)$$ 5. Simplify: $$\det(A) = 30 - 0 - 60 = -30$$ ### Final Answer: $$\boxed{\det(A) = -30}$$ Would you like a breakdown of properties of determinants or applications in solving equations? ### Related Questions: 1. What is the role of the determinant in determining if a matrix is invertible? 2. How does the cofactor expansion method work for determinants? 3. How can determinants be used to find eigenvalues of a matrix? 4. What is the geometric interpretation of a determinant in a 3x3 matrix? 5. How would row reduction affect the computation of a determinant? ### Tip: To simplify determinant calculations for larger matrices, reduce the matrix to row-echelon form or use properties like row/column swaps and scaling.

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Math Problem Analysis

Mathematical Concepts

Matrix Algebra
Determinants

Formulas

Determinant of a 3x3 matrix formula: det(A) = a(ei − fh) − b(di − fg) + c(dh − eg)

Theorems

Properties of Determinants
Expansion of a Determinant Along a Row

Suitable Grade Level

Grades 10-12