What is the determinant of the matrix
(073107300)
? Simplify your answer.
(determinant, matrix, linear algebra, cofactor expansion, square matrix)
Get the complete, step-by-step math solution for: "What is the determinant of the matrix 0 & 7 & 3 1 & 0 & 7 3 & 0 & 0 ? Simplify your answer. (determinant, matrix, linear algebra, cofactor expansion, ...". Powered by SolveForX AI math tutor.
Step-by-Step Solution
Step 1: Choose a row or column for cofactor expansion
To find the determinant of a 3×3 matrix, we can use the cofactor expansion method. This involves choosing a row or a column and then calculating the sum of the products of each element with its corresponding cofactor. It's generally easiest to choose a row or column that contains the most zeros, as this simplifies calculations. In this matrix, the third row and the second column both have two zeros.
Step 2: Expand along the third row
Let's expand along the third row. The elements are a31=3, a32=0, and a33=0. The cofactor for an element aij is (−1)i+j times the determinant of the submatrix obtained by removing the i -th row and j -th column. Since the second and third elements of the third row are zero, their contributions to the determinant will be zero, simplifying the calculation significantly.
Step 3: Calculate the determinant of the 2×2 submatrix
Now we need to calculate the determinant of the 2×2 submatrix: 7037. The determinant of a 2×2 matrix (acbd) is given by ad - bc.
Step 4: Substitute and simplify
Substitute the determinant of the 2×2 submatrix back into the expansion formula. The terms involving zeros will vanish, leaving only the product of 3, 1 (from (−1)3+1), and the determinant of the submatrix.
Step 5: Final calculation
Perform the final multiplication to get the determinant of the matrix.