A determinant of a square matrix is a value calculated from the elements of the matrix.^{[1]} A determinant is defined only for a square matrix. The determinant of matrix A is denoted |A| or det(A).
A determinant is calculated by multiplying the diagonals of the matrix, the adding or subtracting the products of the diagonals. For the 2x2 matrix
multiply the diagonal from the upper left to lower right first (b_{11}·b_{22}). Then multiply the diagonal from the upper right to the lower left (b_{12}·b_{21}). Subtract the second product from the first (b_{11}·b_{22} - b_{12}·b_{21}). This is the determinant of the matrix.
Figure 1 shows how to find the determinant of a 3x3 matrix.. The first diagonals go from top left to bottom right. The numbers multiplied are a_{11}·a_{22}·a_{33}, then a_{12}·a_{23}·a_{31}, then a_{13}·a_{21}·a_{32}. The product of these diagonals are added together.
The diagonals from upper right to bottom left are calculated. The numbers multiplied are a_{13}·a_{22}·a_{31}, then a_{12}·a_{21}·a_{33}, then a_{11}·a_{23}·a_{32}. These products are subtracted from the previous sum.
The function for the determinant of a 3x3 matrix is |A| = a_{11}·a_{22}·a_{33} + a_{12}·a_{23}·a_{31} + a_{13}·a_{21}·a_{32} - a_{13}·a_{22}·a_{31} - a_{12}·a_{21}·a_{33} - a_{11}·a_{23}·a_{32}.
Figure 1: The determinant of a 3x3 matrix |
# | A | B | C | D |
E | F | G | H | I |
J | K | L | M | N |
O | P | Q | R | S |
T | U | V | W | X |
Y | Z |
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