The determinant of a 2 by 2 and a 3 by 3 matrix, minors and cofactors, and expansion along any row or column. Then the properties that make a large determinant easy, and the area of a triangle and the test for three points on a line.
3 lessons, about 1 h 15 min3 simulations10 problems3 practice questionsPython in every lesson
Created by StemPy LabA determinant is a single number that belongs to a square matrix. For a matrix it is . For larger matrices it is found by expanding along a row or a column, using minors and cofactors with the sign pattern . Any row or column gives the same value, so choose the one with the most zeros. A triangular matrix has the product of its diagonal entries as its determinant.
Row and column operations change a determinant in a known way. Swapping two rows changes its sign, multiplying a row by multiplies the determinant by , and adding a multiple of one row to another changes nothing. A determinant with two equal rows, two proportional rows or a row of zeros is . The determinant of a transpose is the same, and . These facts turn a large determinant into zeros and a short expansion.
The area of the triangle with vertices , and is half the absolute value of the determinant whose rows are . Three points are on one line exactly when this determinant is . The same idea gives the equation of the line through two points. A matrix with determinant multiplies every area by .
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