Question

Find the determinant of the matrix. Expand by cofactors on the row or column that appears to make the computations easiest. Use a graphing utility to confirm your result. $$\left[\begin{array}{rrr}1& 4& -2\\ 3& 2& 0\\ -1& 4& 3\end{array}\right]$$

Short answer

The determinant of the given matrix is -92.

Step-by-step solution

Choosing the appropriate row/column for cofactor expansion

Let's choose the second column for cofactor expansion since it has the smallest sum of absolute values.

Determining the Matrix of Minors

Let's compute the minors for the second column. For the first position (element 4), we have to eliminate the first row and second column, which leaves us a 2x2 matrix: $\left[\begin{array}{ccc}3& 0-1& 3\end{array}\right].Thedeterminantofthatminorbecomes\text{(}3\ast 3-0\ast -1=9$. For the second position (element 2), we eliminate the second row and second column, forming a 2x2 matrix: $\left[\begin{array}{ccc}1& -2-1& 3\end{array}\right].Thedeterminantofthatminorbecomes\text{(}1\ast 3--2\ast -1=1$. Lastly for the third position (element 4), we remove the third row and second column resulting into $\left[\begin{array}{ccc}1& -23& 0\end{array}\right].Thedeterminantofthatminoris\text{(}1\ast 0--2\ast 3=6$. Therefore, the matrix of minors is $\left[\begin{array}{ccccccc}9& .& ..& 1& ..& 6& .\end{array}\right]$

Determining the Matrix of Cofactors

The sign of a cofactor is determined by the sum of its row and column number. If the sum is odd, the sign is negative, otherwise, it is positive. So for our matrix, the cofactors for the second column should be: $4\ast (-1{)}^{1+2}\ast 9,2\ast (-1{)}^{2+2}\ast 1,4\ast (-1{)}^{3+2}\ast 6$, thus, our matrix of cofactors becomes $\left[\begin{array}{ccccccc}-36& .& ..& 2& ..& -24& .\end{array}\right]$.

Calculating the Determinant

We can now calculate the determinant of the initial matrix using the cofactors. The determinant is computed as the sum of the product of the elements of any row or column and their corresponding cofactor. Hence, determinant = $1\ast -36+2\ast 2+-24\ast 4=-92$.

Verifying with a Graphing Tool

Verify the computed determinant using a graphing software. Such tool typically provides the function to calculate the determinant of a given matrix. The output from the graphing tool should match our calculated determinant of -92.

Key concepts

Determinant Calculation

Understanding how to calculate the determinant of a matrix is a fundamental skill in linear algebra. The determinant is a scalar value that provides important information about the properties of a matrix. For a 2x2 matrix, the determinant is calculated as the product of the diagonal elements subtracted by the product of the off-diagonal elements. For larger square matrices, one common method to calculate the determinant is by performing a cofactor expansion, which involves creating a matrix of minors, alternating signs accordingly to create a matrix of cofactors, and then summing the products of the elements by their corresponding cofactors along a row or column.

You may find that selecting the row or column with the smallest absolute values or zeros can simplify calculations, as done in the textbook example. This strategic choice can significantly reduce the complexity of the calculations required to find the determinant.

Cofactor Expansion

Cofactor expansion, also known as Laplace's expansion, is a technique used to compute the determinant of a square matrix. It involves expanding the determinant across a single row or column, multiplying each element by its corresponding 'cofactor'. The cofactor of an element is calculated by taking the determinant of a smaller matrix that results from removing the element's row and column, then multiplying by $(-1{)}^{i+j}$ where $i$ and $j$ are the row and column indices of the element. When you choose the best row or column (often one with zeros or smaller values, to make calculations easier), you can minimize the number of operations needed, as shown in the step-by-step solution.

Matrix of Minors

The matrix of minors is a stage in the process of determinant calculation where you compute a minor for each element of a square matrix. A minor is, in essence, the determinant of a smaller matrix obtained by deleting the row and column of the element of interest.

To illustrate, consider each element of a selected row or column. For each, eliminate the respective row and column, then compute the determinant of the resulting smaller matrix. Continue this process for all elements along the chosen row or column. The matrix formed by these minors is known as the 'matrix of minors'.

Matrix of Cofactors

Once we have the matrix of minors, obtaining the matrix of cofactors is the subsequent step. Each element of the matrix of minors is adjusted by a sign based on its position. The rule for the sign is determined by the formula $(-1{)}^{i+j}$ where $i$ and $j$ are the row and column indices, respectively. If the sum of $i+j$ is even, the sign is positive; if it is odd, the sign is negative.

Creating the matrix of cofactors by applying the sign changes to the matrix of minors is a crucial action in the determinant calculation process. As visible in the provided example, maybe not all cofactors are necessary, especially if you have chosen a row or column with zeros, which simplifies the process even more.

Graphing Utility Verification

After manually calculating the determinant of a matrix, it's both reassuring and prudent to verify your result with technology. Graphing utilities, such as calculators or dedicated software, can swiftly confirm determinants with high accuracy.

To check your calculation, simply enter the matrix into the utility and use the built-in function to compute the determinant. The result should coincide with your manual calculation. This step provides an additional layer of confidence in your understanding of the concept and helps you catch any potential errors made during calculations.