Question

Find the inverse of the matrix (if it exists). $$ \left[\begin{array}{rrr} 1 & 1 & 2 \\ 3 & 1 & 0 \\ -2 & 0 & 3 \end{array}\right] $$

Short answer

The inverse of the given matrix is \[\begin{pmatrix}-3/2 & 3/2 & -1\3/2 & -5/2 & 3\1 & 0 & -1\end{pmatrix}\]

Step-by-step solution

Calculate the determinant

First, calculate the determinant of the original matrix, using the formula: \[det(A) = a(ei−fh) − b(di−fg) + c(dh−eg)\]for the matrix \[A=\begin{pmatrix}a&b&c\d&e&f\g&h&i\end{pmatrix}\]Substitute the elements of the given matrix into the formula to get the determinant = \(1*(1*3-0*0) -1*(3*3-(-2)*0)+2*(3*0 - 1*(-2)) = 3 - 9 +4 = -2\]. If determinant equals zero, the matrix has no inverse. In this case, determinant is not zero, so carry on with finding the matrix of minors.

Calculate the matrix of minors

The minor of an element in a matrix is the determinant of the smaller matrix, formed by removing the row and column of that element. The minor for each element in our 3x3 matrix is calculated as follows:\[\begin{pmatrix}(1*3 - 0*0) & (1*3 - 0*2) & (1*1 - 1*3)\(1*3 - 2*0) & (1*3 - 1*(-2)) & (1*1 - 2*1)\(1*0 - 1*2) & (3*2 - 1*0) & (3*1-1*1)\end{pmatrix}= \begin{pmatrix}3 & 3 & -2\3 & 5 & 0\-2 & 6 & 2\end{pmatrix}\]

Calculate the cofactor matrix

The cofactor matrix is found by applying a checkerboard of minuses to the 'Matrix of Minors'. Starting with a plus for the (1,1) position, alternate between plus and minus through the matrix. For the given matrix, after applying the checkerboard of minuses, we obtain:\[\begin{pmatrix}+3 & -3 & -2\-3 & +5 & 0\+2 & -6 & +2\end{pmatrix}\]

Transpose the cofactor matrix

Take the transpose of the cofactor matrix by swapping the element at row i, column j with the element at row j, column i. The transpose is:\[\begin{pmatrix}+3 & -3 & +2\-3 & +5 & -6\-2 & 0 & +2\end{pmatrix}\]

Divide each element by the determinant

Finally, since the determinant is -2, divide each element in the transposed cofactor matrix by -2 to obtain the inverse matrix:\[\begin{pmatrix}-3/2 & 3/2 & -1\3/2 & -5/2 & 3\1 & 0 & -1\end{pmatrix}\]

Key concepts

Determinant Calculation

The determinant is a critical value needed to find whether a matrix has an inverse. For a 3x3 matrix, the determinant is calculated using the formula:\[det(A) = a(ei−fh) − b(di−fg) + c(dh−eg) \]Here, the matrix \(A\) comes with elements \(a, b, c\), etc., representing specific positions. The product of terms in the formula involves multiplying matrix elements, following specific patterns. In the given exercise, after plugging values from the matrix \(\begin{pmatrix}1 & 1 & 2\3 & 1 & 0\-2 & 0 & 3\end{pmatrix}\), the determinant is found as \(-2\). If the result were zero, the matrix would not be invertible. Since we get \(-2\), the process can continue to find the inverse.

Matrix of Minors

The matrix of minors consists of the determinants of 2x2 submatrices. You form these submatrices by excluding the row and column of each element. For each element in the original 3x3 matrix, eliminate its respective row and column to form the smaller matrix, then compute its determinant.For instance:- The minor for element \(a_{11}\) is calculated as \((ei - fh)\).- The minor for element \(a_{12}\) is \((di - fg)\), and so forth.The calculated minors for each element are arranged to create a new matrix, termed the matrix of minors:\[\begin{pmatrix}3 & 3 & -2\3 & 5 & 0\-2 & 6 & 2\end{pmatrix}\]Creating this matrix is key, as it facilitates finding the cofactor matrix next.

Cofactor Matrix

The cofactor matrix is derived by applying a pattern of signs to the matrix of minors. This pattern consists of alternating plus and minus signs, starting with a plus at the top-left.The pattern looks like:- For a \(3x3\) matrix, apply: \((+ - +)\), \((- + -)\), and \((+ - +)\).This results in the elements of the cofactor matrix:\[\begin{pmatrix}+3 & -3 & -2\-3 & +5 & 0\+2 & -6 & +2\end{pmatrix}\]This step plays a crucial role, ensuring the correct orientation and scaling needed for the final steps of matrix inversion.

Transpose of Matrix

Transposing the cofactor matrix involves swapping elements around its diagonal. This means changing the element at row \(i\), column \(j\) to row \(j\), column \(i\).For the cofactor matrix:- Swap elements to get from \(\begin{pmatrix}+3 & -3 & -2\-3 & +5 & 0\+2 & -6 & +2\end{pmatrix}\) to\[\begin{pmatrix}+3 & -3 & +2\-3 & +5 & -6\-2 & 0 & +2\end{pmatrix}\]This transposition aligns the matrix for the final computation step, turning it into the adjugate matrix. The transpose is essential, translating the structured minor-cofactor layout into a form suitable for division by the determinant, ultimately leading to the actual inverse matrix once each element is divided by the determinant value.