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
A matrix is invertible if and only if its determinant is non-zero.
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.
If a matrix is not invertible, it is said to be singular, and its determinant is zero.
Invertible matrices play a crucial role in solving systems of linear equations using the inverse method, as described in section 9.7.
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.
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.
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.
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.