Problem 20 - Algebra and Trigonometry

Question

Find the inverse of the matrix if it exists. $$ \left[\begin{array}{lll}{2} & {1} & {0} \\ {1} & {1} & {4} \\ {2} & {1} & {2}\end{array}\right] $$

Step-by-Step Solution

Answer

The inverse of the matrix is \(\begin{bmatrix} -1 & 0 & -1 \\ 2 & -1 & 4 \\ -0.5 & 1 & -0.5 \end{bmatrix}\).

Step 1: Define the Matrix A

The given matrix is defined as \[A = \begin{bmatrix} 2 & 1 & 0 \ 1 & 1 & 4 \ 2 & 1 & 2 \end{bmatrix}\] Our task is to determine if this matrix has an inverse.

Step 2: Calculate the Determinant of A

The determinant of a 3x3 matrix \(A\) given by \[A = \begin{bmatrix} a & b & c \ d & e & f \ g & h & i \end{bmatrix}\] is calculated as \[det(A) = a(ei - fh) - b(di - fg) + c(dh - eg)\] Substitute the values from \(A\):\[det(A) = 2(1 \times 2 - 4 \times 1) - 1(1 \times 2 - 4 \times 2) + 0(1 \times 1 - 1 \times 2)\]\[det(A) = 2(2 - 4) - 1(2 - 8)\]\[det(A) = 2(-2) - 1(-6)\]\[det(A) = -4 + 6 = 2\] The determinant is 2.

Step 3: Assess Invertibility

A matrix is invertible if its determinant is not zero. Since \(det(A) = 2eq 0\), the matrix has an inverse.

Step 4: Calculate the Adjugate of A

The adjugate of a matrix is the transpose of its cofactor matrix. To find the cofactor matrix, calculate the determinant of each 2x2 minor associated with each element of \(A\), apply the checkerboard pattern of signs \((+,-,+; -,+,-; +,-,+)\), and assemble them.- Cofactor of \(a_{11}\): \((-1)^{1+1} \cdot det\begin{bmatrix}1 & 4\ 1 & 2\end{bmatrix} = 1 \, \Rightarrow \, 1(2 - 4) = -2\)- Cofactor of \(a_{12}\): \((-1)^{1+2} \cdot det\begin{bmatrix}1 & 4\ 2 & 2\end{bmatrix} = -1 \, \Rightarrow \, -1(2 \times 2 - 4 \times 2) = 4\)- Cofactor of \(a_{13}\): \((-1)^{1+3} \cdot det\begin{bmatrix}1 & 1\ 2 & 1\end{bmatrix} = 1 \, \Rightarrow \, 1(1 - 2) = -1\)- Continue similarly for other elements.The cofactor matrix is \[C = \begin{bmatrix} -2 & 4 & -1 \ 0 & -2 & 2 \ -2 & 8 & -1 \end{bmatrix}\] Thus, the adjugate is the transpose \[C^T = \begin{bmatrix} -2 & 0 & -2 \ 4 & -2 & 8 \ -1 & 2 & -1 \end{bmatrix}\]

Step 5: Compute the Inverse Matrix

The inverse of the matrix \(A\) is calculated using the formula \[A^{-1} = \frac{1}{det(A)} \cdot adj(A)\] Substituting the determinant and the adjugate values:\[A^{-1} = \frac{1}{2} \cdot \begin{bmatrix} -2 & 0 & -2 \ 4 & -2 & 8 \ -1 & 2 & -1 \end{bmatrix}\]Perform the scalar multiplication:\[A^{-1} = \begin{bmatrix} -1 & 0 & -1 \ 2 & -1 & 4 \ -0.5 & 1 & -0.5 \end{bmatrix}\] Thus, the inverse matrix is obtained.

Key Concepts

Determinant of a Matrix

The determinant of a matrix is a special number that can be calculated from its elements. For a 3x3 matrix, like the one given in the exercise, the formula for the determinant is:\[\text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg)\]This formula might look daunting, but it's simply about finding a specific combination of the matrix elements. Let's break it down:

• Identify the first element of each row, column, or the principal diagonal when computing its contribution to the determinant.
• Subtract or add the results according to the order in the formula, keeping track of the plus and minus signs.

In our original problem, we calculated: \[\text{det}(A) = 2(-2) - 1(-6) = -4 + 6 = 2\]This result tells us important information about the matrix, particularly regarding its invertibility.

Adjugate of a Matrix

The adjugate, or adjoint, of a matrix is a crucial step in finding a matrix's inverse. It involves creating a new matrix from the cofactors of the original matrix and then transposing it. Here's how you do it:

• First, calculate the cofactors for each element of the original matrix by focusing on the 2x2 minors of the matrix. A minor is formed by removing the row and column of that specific element.
• Apply a checkerboard pattern of signs to the cofactors (+,-,+; -,+,-; +,-,+). This operation changes their values and signs properly.
• Transpose the matrix of cofactors, swapping rows and columns, to get the adjugate matrix.
• For our example, the cofactor matrix was calculated and transposed to create the adjugate matrix:\[adj(A)^T = \begin{bmatrix} -2 & 0 & -2 \ 4 & -2 & 8 \ -1 & 2 & -1 \end{bmatrix}\]

This process may seem complex, but it's just a systematic way to rearrange and manipulate the elements of the matrix.

Cofactor Matrix

The cofactor matrix is built by finding the cofactor of each element in the matrix. Calculating cofactors involves taking each element and considering a smaller, related submatrix.Here's how it works:

• For each element of the 3x3 matrix, create a 2x2 matrix by removing the row and column the element is part of.
• Calculate the determinant of this smaller 2x2 matrix.
• Apply a specific sign to each element, following the checkerboard pattern of signs (+,-,+ from the top left).

For example, the cofactor for the element at the first row and first column (\(a_{11}\)) is calculated using its 2x2 minor, where its position determines its sign.For our example, the cofactor of the first element:\[(-1)^{1+1} \times \text{det}\begin{bmatrix}1 & 4 \ 1 & 2\end{bmatrix} = -2\]Each cofactor becomes an entry in the cofactor matrix, which is the precursor to creating the adjugate.

Invertibility of Matrices

Invertibility is an essential concept in linear algebra. It tells us whether a matrix can be inverted. The key factor in determining this is the determinant of the matrix.A matrix is invertible if its determinant is not zero.

• If \(\text{det}(A) eq 0\), the matrix has an inverse, and we can proceed with further calculations like finding the adjugate.
• If \(\text{det}(A) = 0\), the matrix is not invertible, meaning no inverse exists.

In our example, the determinant was 2, not zero. Therefore, we know that our matrix has an inverse.Understanding this concept is crucial for solving matrix-related problems and emphasizes why calculating the determinant is often the first step when analyzing matrices for invertibility.