The Determinant

Section 3.6 The Determinant

Checkpoint 3.6.1.

Given that for \(T:\mathbb{R}^2 \rightarrow \mathbb{R}^2\text{,}\) \(T(1,0) = (2,3)\) and \(T(0,1) = (4,5)\text{,}\) find a formula for \(T(x_1,x_2)\text{.}\)

Checkpoint 3.6.2.

Let \(f:\mathbb{R}^2 \rightarrow \mathbb{R}^3\) be defined by \(f(x_1,x_2) =(x_1, x_1\cdot x_2,x_2)\text{.}\) Is \(f\) a linear transformation? If possible find a matrix \(\mtx{A}\) such that

\begin{equation*} f(x_1,x_2) = \mtx{A}\left[ \begin{array}{c} x_1 \\ x_2 \end{array} \right] \end{equation*}

Checkpoint 3.6.3.

Let \(g:\mathbb{R}^2 \rightarrow \mathbb{R}^3\) be defined by \(g(x_1,x_2) =(x_1, x_1+x_2,x_2)\text{.}\) Is \(g\) a linear transformation? If possible find a matrix \(\mtx{A}\) such that

\begin{equation*} g(x_1,x_2) = \mtx{A}\left[ \begin{array}{c} x_1 \\ x_2 \end{array} \right] \end{equation*}

Theorem 3.6.4.

A function \(f: \mathbb{R}^m \longrightarrow \mathbb{R}^n\) is a linear transformation if and only if it can be defined as multiplication by a matrix: \(f(\vect{v} ) = \mtx{A}\vect{v}\text{.}\) The size of the matrix will be \(\underline{\hspace{.1in} } \times \underline{\hspace{.1in} }\text{.}\)

To indicate the linear transformation that results from multiplication by the matrix \(\mtx{A}\text{,}\) write \(T_\mtx{A} :\mathbb{R}^m \longrightarrow \mathbb{R}^n\text{.}\)

Challenge 3.6.5.

What must be true about the columns of \(\mtx{A}\) for the linear transformation \(T_\mtx{A}\) to be onto? What must be true about the columns of \(\mtx{A}\) for \(T_\mtx{A}\) to be \(1-1\text{?}\)

To indicate the matrix for a given linear transformation \(T:\mathbb{L}_1 \longrightarrow \mathbb{L}_2\) uses \(\alpha\) as the basis for \(\mathbb{L}_1\) and \(\beta\) as the basis for \(\mathbb{L}_2\) the matrix is written \([T]_{\beta, \alpha}\text{.}\) This means \([T(P)]_{\beta} =[T]_{\beta, \alpha}[P]_{\alpha}\) and \([T_2 \circ T_1]_{\gamma, \alpha}\) and \([T_2]_{\gamma, \beta}[T_1]_{\beta, \alpha}\text{.}\)

Definition 3.6.6.

There is a useful process for assigning a real number, called the determinant, to any square matrix. For \(n\times n\) matrix \(\mtx{A}\) this number is written det(\(\mtx{A}\)) or \(|\mtx{A}|\text{.}\)

The determinant for \(2 \times 2\) matrix \(\mtx{A}= \left[ \begin{array}{cc}a\amp b\\c\amp d \end{array} \right]\) is the real number:

\begin{equation*} \left| \begin{array}{cc}a\amp b\\c\amp d \end{array} \right| = ad-bc \end{equation*}

The determinant for a \(3 \times 3\) matrix is:

\begin{equation*} \left| \begin{array}{ccc}a\amp b\amp c\\d\amp e\amp f\\g\amp h\amp i \end{array} \right| = aei+bfg+cdh-ceg-bdi-afh \end{equation*}

Discussion Question 3.6.7.

There are many ways this sum can be factored, each giving a different way of understanding the determinant of a \(3 \times 3\) matrix, one example is:

\(a(ei-fh)-b(di-fg)+c(dh-eg) = a\left| \begin{array}{cc}e\amp f\\h\amp i \end{array} \right|-b\left| \begin{array}{cc}d\amp f\\g\amp i \end{array} \right| +c\left| \begin{array}{cc}d\amp e\\g\amp h \end{array} \right|\)

Another example is:

\(-b(di-fg) + e(ei-cg) - h(af-cd) = -b\left| \begin{array}{cc}d\amp f\\g\amp i \end{array} \right| + e\left| \begin{array}{cc}a\amp c\\g\amp i \end{array} \right| -h\left| \begin{array}{cc}a\amp c\\d\amp f \end{array} \right|\)

Find another example. What pattern do all of these examples have in common. This pattern can be used to extend the definition of determinant to larger square matrices.