Problem 19 - A Graphical Approach to College Algebra

Question

For each matrix, find \(A^{-1}\) if it exists. Do not use a calculator. $$A=\left[\begin{array}{rrr} 1 & 0 & 1 \\ 2 & 1 & 3 \\ -1 & 1 & 1 \end{array}\right]$$

Step-by-Step Solution

Answer

The inverse of matrix \( A \) is \( \begin{bmatrix} -2 & -2 & 1 \\ -5 & 2 & -1 \\ 3 & 3 & 1 \end{bmatrix} \).

Step 1: Verify Invertibility

To find if a matrix is invertible, its determinant must be non-zero. Calculate \( \det(A) \) using the formula for a 3x3 matrix determinant:\[\det(A) = a(ei-fh) - b(di-fg) + c(dh-eg)\]For matrix \( A \), this gives:\[\det(A) = 1(1 \cdot 1 - 3 \cdot 1) - 0(2 \cdot 1 - 3 \cdot (-1)) + 1(2 \cdot 1 - 1 \cdot (-1)) = 1(-2) + 0 + 1(3) = 1\]Since \( \det(A) = 1 eq 0 \), \( A \) is invertible.

Step 2: Find Co-factors for Matrix Elements

Find the minors and co-factors for each element of \( A \).For example, the cofactor of element \( a_{11} \) (1) is given by:\[ C_{11} = \det \left( \begin{array}{cc} 1 & 3 \ 1 & 1 \end{array} \right) = (1 \cdot 1 - 3 \cdot 1) = -2 \]Repeat for each element of \( A \):- \( C_{12} = -\det \left( \begin{array}{cc} 2 & 3 \ -1 & 1 \end{array} \right) = -(2 \cdot 1 - 3 \cdot (-1)) = -5 \)- \( C_{13} = \det \left( \begin{array}{cc} 2 & 1 \ -1 & 1 \end{array} \right) = (2 \cdot 1 - 1 \cdot (-1)) = 3 \)This process will be repeated for all elements of the matrix \( A \) to populate the cofactor matrix.

Step 3: Construct Cofactor Matrix

The cofactor matrix \( C \) for matrix \( A \) is:\[C = \begin{bmatrix}-2 & -5 & 3 \-2 & 2 & 3 \1 & -1 & 1\end{bmatrix}\]

Step 4: Transpose the Cofactor Matrix

Transpose the cofactor matrix \( C \) to get the adjugate matrix \( \text{adj}(A) \):\[\text{adj}(A) = \begin{bmatrix}-2 & -2 & 1 \-5 & 2 & -1 \3 & 3 & 1\end{bmatrix}\]

Step 5: Multiply by Determinant Inverse

The inverse of \( A \), \( A^{-1} \), is given by:\[A^{-1} = \frac{1}{\det(A)} \cdot \text{adj}(A)\]Since \( \det(A) = 1 \),\[A^{-1} = 1 \cdot \begin{bmatrix}-2 & -2 & 1 \-5 & 2 & -1 \3 & 3 & 1\end{bmatrix} = \begin{bmatrix}-2 & -2 & 1 \-5 & 2 & -1 \3 & 3 & 1\end{bmatrix}\]

Step 6: Confirm Result

Verify the result by multiplying \( A \) and \( A^{-1} \) to see if it results in the identity matrix. This step confirms the calculation.

Key Concepts

Invertibility

To determine whether a matrix is invertible, we first need to check its determinant. A matrix has an inverse if and only if its determinant is non-zero. This rule comes from linear algebra and ensures that the operations required to reverse the matrix are valid.
In the context of our exercise, we calculate the determinant of the given matrix to see if it is truly invertible. If the determinant equals zero, the matrix does not possess an inverse, making it non-invertible.

Determinant

The determinant is a special number that gives us important information about the matrix. For a square matrix, the determinant helps in understanding properties like invertibility and in calculating other relevant matrices such as the cofactor and adjugate matrices.
For a 3x3 matrix, the determinant can be calculated using the formula \[\det(A) = a(ei-fh) - b(di-fg) + c(dh-eg) \]This formula involves a selection of elements from the matrix and their corresponding minors, sequentially multiplied and summed up to obtain the determinant value. In our problem, the determinant is found to be 1, indicating that the matrix is invertible.

Cofactor Matrix

The concept of a cofactor matrix is a cornerstone in the computation of a matrix's inverse. To find it, we need to compute the minor for each element of the matrix and then apply a sign based on the position of the element. For any element in position \(a_{ij}\), this involves calculating the determinant of the smaller matrix formed by deleting the i-th row and j-th column.
Each entry of the cofactor matrix is then given by \((-1)^{i+j} M_{ij}\) where \(M_{ij}\) is the minor for the respective element. The cofactor matrix obtained from this exercise serves as an intermediary in finding further matrix calculations.

Adjugate

The adjugate, also known as the adjoint, is derived by transposing the cofactor matrix. This means flipping the matrix over its diagonal, swapping the row and column indices of each element. The adjugate plays an integral role because when it is multiplied by the scalar reciprocal of the determinant, it directly gives us the inverse of the matrix.
In this step, our exercise demonstrates how a cofactor matrix is transformed into the adjugate matrix, ensuring that each element's position is correctly adjusted to reflect this property.

Identity Matrix

In verifying the correctness of the inverse matrix computation, we use the identity matrix. The identity matrix is a special kind of square matrix where all the elements on the main diagonal are 1, and all other elements are 0.

• In matrix multiplication, any matrix multiplied by the identity matrix results in itself.
• The identity matrix acts as the neutral element in matrix operations, similar to the number 1 in scalar multiplication.

In the final step of our exercise, if multiplying a matrix by its inverse results in the identity matrix, it confirms that our inverse computation was successful. This is a crucial validation step in verifying the solution is correct.