Problem 17 - Algebra 2

Question

Determine whether each matrix has an inverse. If an inverse matrix exists, find it. $$ \left[\begin{array}{ll}{1} & {3} \\ {2} & {0}\end{array}\right] $$

Step-by-Step Solution

Answer

The given matrix has an inverse, which is \[\left[\begin{array}{cc}0 & 1/2\ 1/3 & -1/6\end{array}\right]\].

Step 1: Calculate the Determinant

To determine if an inverse exists for the given matrix, find the determinant. The determinant of a 2x2 matrix \( A = \left[\begin{array}{cc}a & b\ c & d\end{array}\right] \) can be calculated as \( \text{det}(A) = ad - bc \). For the given matrix \( \left[\begin{array}{ll}1 & 3\ 2 & 0\end{array}\right] \) the determinant is \( 1*0 - 2*3 = -6 \) which is not equal to zero.

Step 2: Determine if the Inverse Exists

Since the determinant is non-zero (\( -6 \)), the matrix has an inverse.

Step 3: Calculate the Inverse

The inverse of a 2x2 matrix \( A \) when \( \text{det}(A) \) is not equal to zero can be found using the formula \( A^{-1} = \frac{1}{\text{det}(A)} \left[\begin{array}{cc}d & -b\ -c & a\end{array}\right] \). Apply this to the given matrix: \( \left[\begin{array}{ll}1 & 3\ 2 & 0\end{array}\right]^{-1} = \frac{1}{-6} \left[\begin{array}{cc}0 & -3\ -2 & 1\end{array}\right] \) which simplifies to \( \left[\begin{array}{cc}0 & 1/2\ 1/3 & -1/6\end{array}\right] \).

Key Concepts

Determinant of a Matrix

In matrix algebra, the determinant is a special number that provides valuable information about a matrix. For a 2x2 matrix like
\( A = \left[\begin{array}{cc} a & b \ c & d \end{array}\right] \),
the determinant can be calculated with a simple formula:
\( \text{det}(A) = ad - bc \).
The determinant reflects certain properties of the matrix, such as whether it is invertible or singular, and the scale factor of linear transformations it represents. Moreover, the computation of the determinant is a crucial step in various matrix operations, including finding the inverse of a matrix.

Inverse of a 2x2 Matrix

For a 2x2 matrix, finding the inverse is quite straightforward if the determinant is non-zero. Given a matrix
\( A = \left[\begin{array}{cc} a & b \ c & d \end{array}\right] \),
its inverse is given by:
\( A^{-1} = \frac{1}{\text{det}(A)} \left[\begin{array}{cc} d & -b \ -c & a \end{array}\right] \).
However, it is important to note that this formula is only applicable if the determinant of the matrix (\(ad - bc\)) is not equal to zero. The values \(d\) and \(a\) in the inverse matrix are switched, and the signs of \(b\) and \(c\) are flipped. This process is essential for various applications in linear algebra, such as solving systems of linear equations.

Non-Zero Determinant

When dealing with matrices, it's critical to understand the significance of a non-zero determinant. A non-zero determinant indicates that the matrix is invertible and has an inverse. Conversely, a determinant of zero implies that the matrix is singular, which means it does not have an inverse. In the context of systems of equations, a non-zero determinant suggests that the system has a unique solution. This is an essential concept since an inverse matrix is often used to compute solutions to linear equations or to change the basis in vector spaces.

Matrix Algebra

Matrix algebra comprises various operations that can be performed on matrices, including addition, subtraction, multiplication, and inversion. In usage, it’s akin to performing arithmetic on numbers but with a set of rules tailored for matrices. When solving for the inverse, a key aspect of matrix algebra, one must be aware of the prerequisites, such as the existence of a non-zero determinant. Learning matrix algebra is fundamental to understanding linear transformations, and it has wide-ranging applications in fields such as computer graphics, engineering, and data science.