Determinant of a Matrix

Determinant of a Matrix

Determinant of a Matrix

Properties of Determinants The determinant of a matrix and its transpose are equal, that is |AT| = |A|. If a matrix B results from a matrix A by interchanging two rows (columns) of A, then |B| = −|A|.

Properties of Determinants If two rows (columns) of a matrix A are equal, then |A| = 0. If a row (column) of a matrix A consists entirely of zeros, then |A| = 0.

Properties of Determinants If a matrix B is obtained from a matrix A by multiplying one row (column) of A by a number r , then |B| = r |A|. If a matrix B is obtained from a matrix A by adding a multiple of one row (column) of A to another row (column) of A, then |B| = |A|.

Properties of Determinants The determinant of an n×n upper (lower) triangular matrix is the product of its main diagonal entries. The determinant of an n×n diagonal matrix is the product of its main diagonal entries. |AB| = |A| |B |. |An| = |A|n If A is an n×n matrix and r is a scalar , then |r A| = r n |A|. A is nonsingular if and only if |A| ≠ 0.

Problems Example 1: Evaluate

Problems Example 2: If A and B are 5×5 matrices with |A| = 8 and |B| = 2. Find (i) |A2| (ii) |-A| (iii) |AT B -1| (iv) |2A-1B 4A|.

Problems Example 3: Answer each of the following as True or False. Justify your answer. (i) If A and B are n×n matrices, then |AB | = |BA|. (ii) If A and B are n×n matrices, then |A + B | = |A| + |B |. (iii) If A and B are n×n matrices such that AB2 = In , then A is nonsingular.

Problems Example 4: By reduction to triangular form, evaluate

Problems Example 5: