Determinants Mind Map

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 3 by 3 determinant, expand along the row or column containing the most zeros or simplest numbers.

Properties of determinants are rules that allow simplification by operating on rows or columns without expanding fully.

Mention the property used before simplifying, especially in long-answer determinant simplification questions.

The area of a triangle with vertices (x1, y1), (x2, y2), and (x3, y3) can be found using a determinant formula.

If the final determinant is zero, write that the points are collinear instead of saying the triangle has a negative or invalid area.

The minor Mij of an element aij is the determinant obtained by deleting the ith row and jth column. The cofactor Aij is defined as Aij = (-1)^(i+j) Mij.

Write the sign pattern [[+,-,+],[-,+,-],[+,-,+]] before calculating cofactors of a 3 by 3 matrix.

The adjoint of a square matrix A is the transpose of its cofactor matrix. If |A| ≠ 0, then A inverse exists and A^-1 = adj(A)/|A|.

Before finding cofactors, check |A|. If |A| = 0, stop and state that inverse does not exist.

A system of linear equations can be written as AX = B. If |A| ≠ 0, the unique solution is X = A^-1B.

Always define A, X, and B clearly before applying X = A^-1B; this earns method credit and reduces arrangement errors.

A system of linear equations is consistent if it has at least one solution. It is inconsistent if it has no solution.

When |A| = 0, compare both coefficients and constants or use adj(A)B; do not declare the result from determinant alone.