Extra (optional) exercises – Short Day

These exercises are optional extra material for students who have finished the exercises of the day and have the time and interest for more.

Extra exercise 1: Cramer’s rule

Let \(n\) be a natural number. A system of linear equations is given in matrix form, \(\mathbf A \cdot \mathbf x = \mathbf b\), where \(\mathbf b \in \mathbb{F}^n\), and \(\mathbf x\) is a vector of the variables \(x_1,\dots,x_n \in \Bbb F\). We will assume that \(\mathbf A \in \mathbb F^{n \times n}\) is a square matrix with \(\det(\mathbf A) \neq 0\).

Question a

Let \(k\) be an integer between \(1\) and \(n\), and denote by \(\mathbf X_k\) the matrix obtained by replacing the \(k\)’th column of the identity matrix \(\mathbf I_n\) by \(\mathbf x\). Show that \(\det(\mathbf X_k)=x_k\), meaning the \(k\)’th entry in the vector \(\mathbf x\).

Question b

Again, let \(k\) be an integer between \(1\) and \(n\), and denote by \(\mathbf M_k\) the matrix obtained by replacing the \(k\)’th column of the matrix \(\mathbf A\) by \(\mathbf b\). Show that \(\mathbf A \cdot \mathbf x = \mathbf b\) implies \(\mathbf A \cdot \mathbf X_k= \mathbf M_k\).

Question c

Show that \(x_k\) satisfies \(x_k=\det(\mathbf M_k)/\det(\mathbf A)\).

This method for finding the solution to a system of linear equations with an invertible coefficient matrix is known as Cramer’s rule.

Hint

Calculate the determinant of \(\mathbf M_k\).