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. $$ \left[\begin{array}{rrrr} 1 & 4 & 3 & 2 \\ -5 & 6 & 2 & 1 \\ 0 & 0 & 0 & 0 \\ 3 & -2 & 1 & 5 \end{array}\right] $$

Short answer

The determinant of the given matrix is 0.

Step-by-step solution

Select the row or column

Since the third row of this matrix contains all zeros, select the third row to perform the expansion.

Identify Matrix Coefficients and Apply Expansion

Each element in this row can be represented as \(a_{ij}\), where 'i' refers to the row number and 'j' refers to the column number. The determinant of a 4x4 matrix can be found by expanding it as:\(Det(A)=\sum_{j=1}^{4} (-1)^{i+j} a_{ij} M_{ij}\)where \(M_{ij}\) is the minor of the \(a_{ij}\) element, which is the determinant of the (4-1)x(4-1) matrix that results from deleting row 'i' and column 'j'. However, since the selected row contains all zeros, the determinant of the given matrix is 0, regardless of the values of its minors.

Check with Graphing Utility

A graphing utility can be used to find the determinant of this matrix, and it should confirm that the result is 0.

Key concepts

Cofactor Expansion

Cofactor expansion, also known as Laplace's expansion, is a technique used to calculate the determinant of a matrix. It's particularly useful for larger matrices, where direct computation can be cumbersome. The process involves selecting any row or column (preferably one that simplifies the calculation) and expanding along that line. In our example, the third row is chosen because it has all zeros, which makes the calculations much simpler. Each element in the chosen row or column is multiplied by its corresponding cofactor, which is the determinant of the matrix that remains after removing the row and column of that element, prefaced by a plus or minus sign determined by the element's position. The formula for cofactor expansion looks like this:
\[Det(A) = sum_{{j=1}}^{{n}} (-1)^{{i+j}} a_{{ij}} M_{{ij}}\]
where \(Det(A)\) is the determinant of matrix \(A\), \(a_{{ij}}\) is an element of the matrix, and \(M_{{ij}}\) is the cofactor. For a matrix with a row or column of zeros, the determinant is immediately known to be 0, as each term in the expansion will be zero.

Matrix Coefficients

Matrix coefficients are the individual elements that make up a matrix, usually denoted as \(a_{{ij}}\), where \(i\) indicates the row and \(j\) the column in which the element is found. These coefficients play a crucial role in many matrix operations, including the determinant calculation. When performing cofactor expansion, each coefficient is paired with its respective minor, which involves computing the determinant of a smaller matrix. This smaller matrix, or submatrix, is derived by excluding the row and column that contain the coefficient in question. For example, if we were not using the zero row in our problem, we would calculate the determinant of a 3x3 matrix for each element, considering the signs according to their positions. Despite being a straightforward concept, careful attention is needed to correctly sign the minors since the alternating signs can impact the result significantly. Graphing utilities usually handle these coefficients and signs automatically when calculating the determinant.

Graphing Utility Confirmation

Confirming calculation results using a graphing utility is an excellent step to ensure the accuracy of a manually obtained solution. Most graphing calculators or computer algebra systems can compute the determinant of a matrix with a simple command. In our exercise, after manually determining that the determinant is zero using cofactor expansion, we can input the matrix into a graphing utility to check our work. This cross-verification is not only reassuring but also demonstrates the utility of modern tools in solving mathematical problems. It's important for students to familiarize themselves with such technologies, as they can save time and reduce human error in longer, more complicated calculations. However, one should always understand the underlying principles, like cofactor expansion, to grasp what the utility is computing in the background.