Question

Define submatrix and subdeterminant.

Short answer

A submatrix is formed by deleting any row and column from a given matrix, maintaining the relative position of the remaining elements. A subdeterminant is the determinant of a submatrix. To calculate the subdeterminant, find the submatrix by removing a specific row and column, then compute the determinant of the submatrix. For example, with matrix \(A = \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{pmatrix}\), the submatrix \(A_{22} =\begin{pmatrix} 1 & 3 \\ 7 & 9 \end{pmatrix}\), and the subdeterminant is \(\text{det}(A_{22}) = -12\).

Step-by-step solution

Define a matrix

A matrix is a rectangular array of numbers arranged in rows and columns. The entries in the matrix are called elements, and the size of a matrix is described by identifying the number of rows and columns. For example, a 3x3 matrix has 3 rows and 3 columns.

Define a determinant

The determinant is a scalar value that can be computed only from a square matrix (i.e., the number of rows is equal to the number of columns). It is a special number that helps in solving systems of linear equations and finding the inverse of a matrix. The determinant of a 2x2 matrix \(A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}\) is given by \(\text{det}(A) = ad - bc\).

Define a submatrix

A submatrix is formed by deleting any row and column from a given matrix. For instance, given a matrix \(A\), you can create a submatrix by removing the \(i^{\text{th}}\) row and the \(j^{\text{th}}\) column. In other words, a submatrix is a matrix that consists of a subset of the elements of the original matrix but still maintains the relative position of those elements.

Define a subdeterminant

A subdeterminant is the determinant of a submatrix. To calculate the subdeterminant, first, find the submatrix by removing a specific row and column from the original matrix. Then, compute the determinant of the submatrix.

Example of submatrix and subdeterminant calculation

Let's consider a 3x3 matrix \(A = \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{pmatrix}\). To find the submatrix \(A_{22}\) (submatrix formed by deleting the 2nd row and 2nd column), we get: \(A_{22} =\begin{pmatrix} 1 & 3 \\ 7 & 9 \end{pmatrix}\) Now, we can find the subdeterminant, which is the determinant of the submatrix. \(\text{det}(A_{22}) = (1)(9) - (3)(7) = 9 - 21 = -12\) Thus, the subdeterminant of matrix \(A\) when removing the 2nd row and the 2nd column is -12.