Matrix operations

Definition

Matrix operations involve procedures such as addition, subtraction, and multiplication of matrices, which are rectangular arrays of numbers. These operations are fundamental tools for solving systems of linear equations.

5 Must Know Facts For Your Next Test

1. Matrix addition and subtraction require matrices to have the same dimensions.
2. Matrix multiplication is not commutative; that is, $A \times B \neq B \times A$ in general.
3. The identity matrix acts as a multiplicative identity in matrix multiplication, similar to how 1 acts for real numbers.
4. The determinant of a square matrix can be used to determine if the matrix is invertible; a zero determinant means the matrix is not invertible.
5. The inverse of a matrix, if it exists, can be used to solve systems of linear equations through methods like Gaussian elimination.

Review Questions

• What conditions must be met for two matrices to be added or subtracted?
• Why is matrix multiplication not commutative?
• How do you determine if a square matrix has an inverse?

Related terms

Determinant:

A scalar value derived from a square matrix that indicates whether the matrix is invertible.

Identity Matrix:

A square matrix with ones on the diagonal and zeros elsewhere; it serves as the multiplicative identity in matrix operations.

Inverse Matrix:

A matrix that, when multiplied by its original matrix, yields the identity matrix. It exists only if the original matrix has a non-zero determinant.