Determinant of a Matrix

Check that the matrix is square. For order 2, apply ad - bc. For order 3, choose a convenient row or column. Write each cofactor with its sign. Simplify carefully and state the determinant value.

No diagram is required because the concept is algebraic. A boxed 2 by 2 or 3 by 3 determinant layout can help students track diagonal products and cofactors.

Writing ad + bc instead of ad - bc for a 2 by 2 determinant.

The determinant of a square matrix is a scalar value denoted by |A| or det(A). It is defined for square matrices only and can be evaluated by direct formula for order 1 and 2, and by expansion along any row or column for order 3.

The determinant of a square matrix is a scalar value denoted by |A| or det(A). It is defined for square matrices only and can be evaluated by direct formula for order 1 and 2, and by expansion along any row or column for order 3. For [[a,b],[c,d]], determinant = ad - bc. For a 3 by 3 determinant, |A| = a11A11 + a12A12 + a13A13 when expanding along the first row, where Aij is the cofactor of aij. For A = [[2,3],[5,7]], |A| = 2(7) - 3(5) = 14 - 15 = -1.

For a 1 by 1 matrix [a], the determinant is a. For a 2 by 2 matrix, multiply the main diagonal entries and subtract the product of the other diagonal entries. For a 3 by 3 matrix, choose a row or column, multiply each entry by its cofactor, and add the signed terms. The value remains the same whichever row or column is chosen, though some choices make calculation shorter. For [[a,b],[c,d]], determinant = ad - bc. For a 3 by 3 determinant, |A| = a11A11 + a12A12 + a13A13 when expanding along the first row, where Aij is the cofactor of aij. Evaluate |A| for A = [[1,2,3],[0,4,5],[0,0,6]]. Expanding along the first column: |A| = 1×|[[4,5],[0,6]]| - 0 + 0 = 1(4×6 - 5×0) = 24. Therefore, det(A) = 24. Students may be asked to evaluate a determinant directly, simplify a determinant containing variables, or identify whether a determinant is zero. Writing ad + bc instead of ad - bc for a 2 by 2 determinant.