Definition:

2 x 2 Matrices

3 x 3 Matrices

4 x 4 Matrices

## Example (2 x 2 Matrix)

A = [4, 2 ; -3 , 1]

det(A) = (4)*(1) - (2)*(-3)

      = 4-(-6)
= 10


## Example (3 x 3 Matrix)

A = [2, 5, 1 ; 3, 2, 2 ; 1, 5, 2]

det(A) = 2*[(2)*(2) - (2)(5)] - 5*[(3)*(2) - (2)*(1)] + 1*[(3)*(5) - (2)*(1)]

      = 2*(4 - 10) - 5*(6 - 2) + (15 - 2)
= 2*(-6) - 5*(4) + 13
= -12 - 20 + 13
= -19


## Example (4 x 4 Matrix)

A = [2, 3, 7, 2 ; 1, 6, 2, 3 ; 1, 1, 4, 3 ; 4, 6, 5, 1]

det(A) = 2*{6*[(4*1) - (3)*(5)] - 2*[(1)*(1) - (3)*(6)] + 3*[(1)*(5) - ((4)*(6)]} - 3*{1*[(4)*(1) - (3)*(5)] - 2*[(1)*(1) - (3)*(4)] + 3*[(1)*(5) - (4)*(4)]} + 7*{1*[(1)*(1) - (3)*(6)] - 6*[(1)*(1) - (3)*(4)] + 3*[(1)*(6) - (1)*(4)]} - 2*{1*[(1)*(5) - (4)*(6)] - 6*[(1)*(5) - (4)*(4)] + 2*[(1)*(6) - (1)*(4)]}

## Important Properties of the Determinant

- The determinant of the transpose of a matrix will be the same as the determinant of the original matrix.

- If two rows in the matrix are interchanged, the determinant will be the negative of the determinant of the original matrix.

- If two rows or columns in a matrix are the same, the determinant will be zero.

- If there is a row of zeros in the matrix, the determinant will be zero.

- If a row or column in a matrix was multiplied by a real number, then the determinant will be that real number multiplied by the determinant of the original matrix.

- If the matrix is either an upper or a lower triangular matrix, the determinant will be the product of the elements on the main diagonal.

- If two matrices are n x n, then the determinant of one multiplied by the determinant of the other is the determinant of the full matrix of the two next to each other.

## Nonsingular and Similar Matrices

Definition: A square matrix is called nonsingular if its determinant is not equal to zero.

- For square matrices, then the linear combination of the matrix has a nontrivial solution if and only if the matrix is NOT nonsingular.

- For nonsingular matrices, the determinant of the inverse of that matrix is equal to 1 over the determinant of the matrix.

- If two matrices are similar matrices, then the determinants of the two are equal.