Find the determinant of the following matrix: \n\[ A = \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix} \]
Answer
Step 5: Final simplification \n\[ \text{{det}}(A) = -3+12-9 \]
Step 1 ：Step 1: Apply the formula for the determinant of a 3x3 matrix which is given by \n\[ \text{{det}}(A) = a(ei−fh)−b(di−fg)+c(dh−eg) \] where a, b, c, d, e, f, g, h, i are the elements of the matrix.
Step 2 ：Step 2: Substitute the elements of the matrix into the formula \n\[ \text{{det}}(A) = 1(5*9−6*8)−2(4*9−6*7)+3(4*8−5*7) \]
Step 3 ：Step 3: Simplify the expression \n\[ \text{{det}}(A) = 1(45−48)−2(36−42)+3(32−35) \]
Step 4 ：Step 4: Further simplify the expression \n\[ \text{{det}}(A) = 1(-3)−2(-6)+3(-3) \]
Step 5 ：Step 5: Final simplification \n\[ \text{{det}}(A) = -3+12-9 \]
