Invertible Matrix

Definition

An invertible matrix is a square matrix that has an inverse matrix. In other words, it is a matrix that can be multiplied by another matrix to produce the identity matrix, which means that the original matrix can be 'undone' or reversed.

5 Must Know Facts For Your Next Test

1. A matrix is invertible if and only if its determinant is non-zero.
2. The inverse of an invertible matrix $A$ is denoted as $A^{-1}$, and satisfies the equation $A^{-1}A = AA^{-1} = I$, where $I$ is the identity matrix.
3. If a matrix is not invertible, it is said to be singular, and its determinant is zero.
4. Invertible matrices play a crucial role in solving systems of linear equations using the inverse method, as described in section 9.7.
5. The inverse of a 2x2 matrix $\begin{bmatrix} a & b \\ c & d \end{bmatrix}$ is $\frac{1}{ad-bc} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}$.

Review Questions

• Explain the relationship between the determinant of a matrix and its invertibility.
  • The determinant of a matrix is a key property that determines whether the matrix is invertible or not. A matrix is invertible if and only if its determinant is non-zero. If the determinant is zero, the matrix is said to be singular and it does not have an inverse. The determinant can be thought of as a measure of the 'size' or 'volume' of the matrix, and a non-zero determinant indicates that the matrix can be 'undone' or reversed.
• Describe how invertible matrices are used to solve systems of linear equations, as discussed in section 9.7.
  • In section 9.7, the concept of invertible matrices is used to solve systems of linear equations. If a system of linear equations can be represented in matrix form as $Ax = b$, where $A$ is the coefficient matrix, $x$ is the vector of unknowns, and $b$ is the vector of constants, then the system can be solved by multiplying both sides by the inverse of $A$, denoted as $A^{-1}$. This gives $x = A^{-1}b$, which provides the unique solution to the system, provided that $A$ is invertible (i.e., its determinant is non-zero).
• Explain how to find the inverse of a 2x2 matrix, and discuss the significance of this result.
  • The inverse of a 2x2 matrix $\begin{bmatrix} a & b \\ c & d \end{bmatrix}$ can be calculated using the formula $\frac{1}{ad-bc} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix}$. This formula shows that the inverse of a 2x2 matrix depends on the determinant of the matrix, $ad-bc$. If this determinant is non-zero, then the matrix is invertible, and the inverse can be used to solve systems of linear equations involving 2 variables. The ability to easily compute the inverse of a 2x2 matrix is an important result that simplifies the process of solving systems of linear equations in this context.

Related terms

Identity Matrix: The identity matrix is a square matrix with 1s on the main diagonal and 0s elsewhere. When multiplied by another matrix, it leaves that matrix unchanged.

Determinant: The determinant of a matrix is a scalar value that is a function of the entries of the matrix. A matrix is invertible if and only if its determinant is non-zero.

Matrix Inverse: The inverse of a matrix is another matrix that, when multiplied by the original matrix, produces the identity matrix. An invertible matrix has a unique inverse.