Problem 18 - Algebra 2

Question

Determine whether each matrix has an inverse. If an inverse matrix exists, find it. $$ \left[\begin{array}{rr}{6} & {-8} \\ {-3} & {4}\end{array}\right] $$

Step-by-Step Solution

Answer

The determinant is \(24 - (-24) = 48\), which is non-zero, so the inverse exists. The inverse of the matrix is \(\frac{1}{48} \times \begin{bmatrix} 4 & 8 \-3 & 6 \ \/bmatrix}\) or \(\begin{bmatrix} 1/12 & 1/6 \-1/16 & 1/8 \ \/bmatrix}\).

Step 1 - Calculate the Determinant

Compute the determinant of the matrix. The determinant of a 2x2 matrix with elements a, b, c, d is ad - bc. For our matrix, the determinant is calculated as \(6 \times 4 - (-8) \times (-3)\).

Step 2 - Determine if Inverse Exists

Determine if the matrix has an inverse. A matrix has an inverse if and only if its determinant is not equal to zero. Substitute the values we found into the determinant formula to evaluate it.

Step 3 - Using the Determinant to Find the Inverse

If the determinant is non-zero, we can find the inverse of the matrix. The formula to find the inverse of a 2x2 matrix is \(\frac{1}{det(A)} \times \begin{bmatrix} d & -b \-c & a \ \/bmatrix}\), where det(A) is the determinant of the matrix A and a, b, c, d are the elements of matrix A. Replace the det(A) with the calculated determinant and a, b, c, d with the respective elements of the given matrix.

Key Concepts

Determinant of a Matrix

Understanding the determinant of a matrix is crucial in the world of matrix algebra. For a 2x2 matrix, the determinant provides us with an important piece of information that reflects the matrix's invertibility. Specifically, consider a 2x2 matrix \[\left[\begin{array}{cc}a & b\ c & d\end{array}\right]\]. The determinant of this matrix is computed as the product of the entries on the main diagonal subtracted by the product of the entries on the off diagonal, which gives us ad - bc.

This value can be seen as a scaling factor that the matrix applies to area or volume when it transforms geometric shapes. If the determinant is zero, the transformation squashes the space into a lower dimension, which indicates the matrix cannot be inverted as it does not preserve the geometry of space uniquely. If it is non-zero, it means the matrix can be inverted, and an inverse matrix that can undo the transformation exists.

Inverse of a 2x2 Matrix

The inverse of a matrix is akin to the 'undo' function for matrix transformations. For a 2x2 matrix, finding the inverse is relatively straightforward if the determinant is non-zero. The formula to get the inverse uses the determinant and rearranges the entries of the matrix. Given a matrix \[\left[\begin{array}{cc}a & b\ c & d\end{array}\right]\], its inverse is given by \(\frac{1}{det(A)} \times \begin{bmatrix} d & -b \ -c & a \end{bmatrix}\), where \(det(A)\) is the determinant.

This formula swaps the positions of your 'a' and 'd', negates 'b' and 'c', and then multiplies each element by the reciprocal of the determinant. Such a method ensures that when you multiply the original matrix by its inverse, you get the identity matrix, confirming the property that a matrix multiplied by its inverse yields the identity.

Matrix Algebra

Matrix algebra is a cornerstone of linear algebra and encapsulates operations such as addition, subtraction, multiplication, and finding the inverse of matrices. These operations open the door to solving systems of linear equations, transforming geometric objects, and more. In the context of our problem, multiplication is key since we multiply the reciprocal of the matrix's determinant by the adjusted matrix to find its inverse.

It is important to note that matrix multiplication is not commutative—it matters which matrix is first. Additionally, not all matrices can be inverted. Those that can are labeled 'invertible' or 'nonsingular,' and their property of having an inverse allows for solving matrix equations where we seek an unknown matrix that completes an equation.

Properties of Matrices

Matrices come with their own set of properties that govern how they can be manipulated and interpreted. Some key properties include the aforementioned determinant and invertibility, but also the concepts of the identity matrix, transposition, and eigenvectors/values.

The identity matrix acts as the neutral element in matrix multiplication, analogous to '1' in regular multiplication. Any matrix multiplied by the identity retains its original form. The transpose of a matrix is another important concept, created by flipping the matrix over its diagonal, essentially swapping rows with columns. Eigenvectors and eigenvalues give insights into matrix transformations and are essential in many applications, including stability analysis and principal component analysis. Understanding these properties helps in grasping the deeper implications of matrix operations in mathematical and real-world contexts.