Question

Evaluate determinant by calculator or by minors. $$\left|\begin{array}{rrr}1 & 5 & 4 \\\\-3 & 6 & -2 \\\\-1 & 5 & 3\end{array}\right|$$

Short answer

\(|A| = 1 \cdot (6 \cdot 3 - (-2) \cdot 5) - 5 \cdot ((-3) \cdot 3 - (-2)\cdot (-1)) + 4\cdot ((-3) \cdot 5 - 6 \cdot (-1)) = 18 + 10 + 15 + 10 - 120 = -67.\)

Step-by-step solution

Expand the determinant using the first row

The determinant of a 3x3 matrix can be evaluated by expanding along any row or column. We choose the first row for expansion. For matrix \[ \begin{bmatrix} 1 & 5 & 4 \ -3 & 6 & -2 \ -1 & 5 & 3 \end{bmatrix} \], its determinant is \(|A| = a_{11} \cdot C_{11} + a_{12} \cdot C_{12} + a_{13} \cdot C_{13}|\), where \(a_{ij}\) is the element in the ith row and jth column and \(C_{ij}\) is the cofactor of \(a_{ij}\).

Calculate the minors and cofactors for the first row

Calculate the minors for the first row elements. For \(a_{11} = 1\), the minor is \(M_{11} = |\begin{bmatrix} 6 & -2\ 5 & 3\end{bmatrix}|\). For \(a_{12} = 5\), the minor is \(M_{12} = |\begin{bmatrix} -3 & -2\ -1 & 3\end{bmatrix}|\). For \(a_{13} = 4\), the minor is \(M_{13} = |\begin{bmatrix} -3 & 6\ -1 & 5\end{bmatrix}|\). The cofactor \(C_{ij}\) is \(C_{ij} = (-1)^{i+j} \cdot M_{ij}\).

Compute the determinant using the first row expansion

The determinant of the matrix is then \(|A| = a_{11} \cdot C_{11} + a_{12} \cdot C_{12} + a_{13} \cdot C_{13}\). Replace \(a_{ij}\) with their respective elements and cofactors calculated in Step 2 to find the determinant value.

Key concepts

Determinant of a Matrix

The determinant of a matrix is a special scalar value that can be computed from its elements and provides significant insight into the properties of the matrix. For example, it can tell us whether a matrix is invertible, what the volume of the matrix is, and so on. In the case of a 3x3 matrix, such as
\[\begin{pmatrix}1 & 5 & 4 \-3 & 6 & -2 \-1 & 5 & 3\end{pmatrix}\]
the determinant can be found using various methods, including cofactor expansion, which we leverage in this example. The determinant is symbolically denoted by enclosing the matrix in vertical bars, like this: \(|A|\) or \(\det(A)\). Determinants are not only useful in theory but also play a key role in solving linear equations systems and in transformations in geometry.

Cofactor Expansion

Cofactor expansion, also known as Laplace's expansion, is a method used to calculate the determinants of larger matrices. It involves breaking down a matrix into smaller parts, which are easier to manage. Each element \(a_{ij}\) of the matrix is associated with its cofactor \(C_{ij}\), which is calculated as \(C_{ij} = (-1)^{i+j} \cdot M_{ij}\), where \(M_{ij}\) is the minor of the element \(a_{ij}\).

The sign of the cofactor is determined by \((-1)^{i+j}\), which alternates with the position of the element in the matrix. The cofactor expansion allows us to express the determinant as a sum of products of matrix elements and their corresponding cofactors.

Minors in Matrices

Minors are closely associated with the concept of determinant. A minor, denoted as \(M_{ij}\), is the determinant of the submatrix that is left after removing the ith row and jth column from the original matrix. Minors are used to compute the cofactors, which in turn help in evaluating the determinant of the original matrix.

For example, to calculate the minor \(M_{11}\) for our 3x3 matrix, you would remove the first row and the first column, leaving you with the submatrix
\[\begin{pmatrix}6 & -2 \5 & 3\end{pmatrix}\]
and then calculate its determinant. Minors represent smaller, more manageable pieces of the puzzle when computing the overall determinant of a matrix.

Matrix Algebra

Matrix algebra encompasses operations and properties of matrices, such as addition, subtraction, multiplication, and calculation of the determinant and inverse. One aspect of matrix algebra is understanding how the determinant behaves with respect to these operations. For instance, if two square matrices are multiplied, the determinant of the resulting matrix is the product of the determinants of the two original matrices. Moreover, only square matrices have determinants, reflecting the geometric interpretability such as area or volume transformation.

In the context of the given exercise, matrix algebra underpins the process of cofactor expansion to evaluate the determinant. The knowledge of how to correctly multiply elements and the application of the sign pattern for cofactors are all governed by the rules of matrix algebra.