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. $$ \left[\begin{array}{rrr} 2 & -1 & 3 \\ 1 & 4 & 4 \\ 1 & 0 & 2 \end{array}\right] $$

Short answer

The determinant of the matrix is 4.

Step-by-step solution

Choose the row or column

For this matrix, all rows and columns have similar complexity, thus we can expand along any. Let's choose to expand along the first row, so the determinant can be calculated as the sum of the products of each element and their corresponding cofactor, with alternating signs. Our chosen row elements are \(2\), \(-1\) and \(3\). The signs for the elements are \(+ - +\) respectively.

Calculate Cofactors

Next, calculate the cofactors for each element. The cofactor is the determinant of the sub-matrix made by removing the row and column of the chosen element. For the element \(2\), the sub-matrix is \(\begin{bmatrix} 4 & 4 \ 0 & 2\end{bmatrix}\) with cofactor \(4*2 - 4*0 = 8\). For \(-1\), the sub-matrix is \(\begin{bmatrix} 1 & 4 \ 1 & 2\end{bmatrix}\) with cofactor \(1*2 - 4*1 = -2\). For \(3\), the sub-matrix is \(\begin{bmatrix} 1 & 4 \ 1 & 0\end{bmatrix}\) with cofactor \(1*0 - 4*1 = -4\).

Compute the determinant

Now compute the determinant by summing the products of the first row elements and their cofactors with the respective signs: \(2*8 - 1*(-2) + 3*(-4) = 16 -12 = 4.\)

Key concepts

Cofactor Expansion

Cofactor expansion is a method used to calculate the determinant of a square matrix. To understand this process, let's break it down. In cofactor expansion, you choose either a row or a column from the matrix to "expand". Once expanded, each element from that row or column is associated with a cofactor. The cofactor is obtained from the sub-matrix that remains after removing the row and column of the chosen element. Each element and its cofactor are multiplied together, with the product being affected by a checkered pattern of positive and negative signs. These signs alternate across the row or column – starting with a positive sign.

For example, if expanding along the first row consisting of elements 2, -1, and 3, the signs are +, -, + respectively. This means that the first element (2) retains a positive sign, the second element (-1) changes its sign when calculating its contribution, and the third element (3) keeps its positive sign. By summing up these products, you get the determinant of the entire matrix.

Sub-Matrix

Sub-matrix plays a crucial role in finding determinants through cofactor expansion. A sub-matrix is formed by removing one row and one column from the original matrix. The resulting smaller matrix is used to compute the cofactor associated with a particular element.

When calculating the determinant, each element of the chosen row or column is paired with a sub-matrix. For instance, when considering the element 2 from the first row of our matrix, you remove the entire first row and the first column to form a sub-matrix:

• Matrix: \[\begin{bmatrix} 4 & 4 \ 0 & 2 \end{bmatrix}\]

To find the cofactor of 2, calculate the determinant of this sub-matrix, which yields 8. Each element's cofactor is determined by a similar sub-matrix and leads to the complete determinant calculation. Understanding sub-matrices thus simplifies the cofactor method by breaking down the problem into smaller, more manageable parts.

Graphing Utility Verification

Using a graphing utility to verify the determinant calculation is a useful step, particularly for larger matrices where manual calculations become cumbersome. Graphing utilities, generally available in graphing calculators or specific computer applications, allow you to input a matrix and automatically compute its determinant.

In our example, after calculating the determinant manually through cofactor expansion, which resulted in a determinant of 4, using a graphing utility provides a quick and reliable check. By entering the matrix into the utility, you can ensure that the manually calculated determinant matches the result the tool provides. One advantage of using a graphing utility is the confidence it provides in confirming your manual solution, especially useful in educational or testing environments. It acts as both a learning aid and a backup to prevent errors due to miscalculations or misunderstanding the cofactor process. This step of verification also helps reinforce your understanding of determinants by showing the correlation between manual calculations and automated assistance.