Problem 18 - College Algebra

Question

use the fact that if \(A=\left[\begin{array}{ll}a & b \\ c & d\end{array}\right],\) then \(A^{-1}=\frac{1}{a d-b c}\left[\begin{array}{rr}d & -b \\ -c & a\end{array}\right]\) to find the inverse of each matrix, if possible. Check that \(A A^{-1}=I_{2}\) and \(A^{-1} A=I_{2}\) $$ A=\left[\begin{array}{rr} 6 & -3 \\ -2 & 1 \end{array}\right] $$

Step-by-Step Solution

Answer

The given matrix does not have an inverse due to the fact that its determinant equals to zero.

Step 1: Application of Inverse Matrix Formula

Using the given formula for the inverse of a 2x2 matrix: \(A^{-1}=\frac{1}{a d-b c}\left[\begin{array}{rr}d & -b \ -c & a\end{array}\right]\), substitute \(a=6\), \(b=-3\), \(c=-2\), and \(d=1\) into the formula.

Step 2: Calculate the Denominator of the Fraction

Calculate the denominator part first, which is the determinant of matrix A: \(a d-b c\). In other words, \(6*1 - -3*-2 = 6 - 6 = 0\). Note that if the result is '0', it implies the matrix does not have inverse. If not '0', continue for the calculation.

Step 3: Calculate the Numerator of the Fraction

If the determinant is non-zero, continue to find the numerator of the fraction: \(d=-b\ -c=a\), which gives us the result \(\left[\begin{array}{rr}1 & 3 \ 2 & 6\end{array}\right]\).

Step 4: Compose the Inverse Matrix

The numerator is divided by the denominator calculated in previous step to get the inverse: \(A^{-1}\) = \(\frac{1}{0}\left[\begin{array}{rr}1 & 3 \ 2 & 6\end{array}\right]\). However in this case, we cannot compute the inverse matrix due to division by zero.

Step 5: Verify the Inverse Matrix

Since the given matrix A doesn't have an inverse, there is no need to verify it using the multiplication \(A A^{-1}=I_{2}\) and \(A^{-1} A=I_{2}\).

Key Concepts

Determinant

The determinant is a special value that you can calculate from a square matrix. It provides essential insight into the matrix, particularly whether it is invertible or not. For a 2x2 matrix, which is arranged as follows: \[ \begin{bmatrix} a & b \ c & d \end{bmatrix} \] the determinant is calculated using the formula \( ad - bc \). This is a straightforward operation, involving only multiplication and subtraction.

When the determinant equals zero, it signals that the matrix doesn't have an inverse, which means it's singular. Conversely, if the determinant is not zero, the matrix is invertible, and its inverse can be found. This "non-zero determinant" condition is crucial as it determines the matrix's invertibility.

2x2 Matrix

A 2x2 matrix is one of the simplest forms of matrices. It has two rows and two columns, making it a square matrix.

These matrices are often used in various mathematical computations and real-life applications, such as transforming coordinates in geometry or encoding data. The general form of a 2x2 matrix looks like this:\[ \begin{bmatrix} a & b \ c & d \end{bmatrix} \]

• "a" and "d" are the elements in the main diagonal.
• "b" and "c" are the off-diagonal elements.

The role of each element is vital as they directly impact calculations, such as determining the determinant or finding the inverse.

Matrix Multiplication

Matrix multiplication is a fundamental operation where you multiply two matrices. To multiply two matrices, the number of columns in the first matrix must equal the number of rows in the second matrix. For example, multiplying a 2x2 matrix with another 2x2 matrix is permissible.

When multiplying two matrices, you calculate each element of the resulting matrix by:

• Taking the dot product of rows from the first matrix and columns from the second matrix.
• Adding up the products obtained from multiplying corresponding elements.

Matrix multiplication is critical for many operations, including testing if you have calculated an inverse correctly. For an inverse, these operations must yield an identity matrix when multiplied by the original matrix.

Identity Matrix

An identity matrix is like the number one in the matrix world. It is a special type of square matrix where all elements are zero, except those on the main diagonal, which are all ones.

For a 2x2 matrix, the identity matrix looks like this:\[ \begin{bmatrix} 1 & 0 \0 & 1 \end{bmatrix} \]This matrix plays a crucial role in matrix multiplication, as multiplying any matrix by its inverse must result in an identity matrix. This verification step ensures that you have correctly computed the inverse of a matrix. If \( A \) is your matrix and \( A^{-1} \) is its inverse, then \( A \times A^{-1} \) will equal the identity matrix if the inverse is correctly found.