Matrix's

**mr_motivator** · February 17th 2009, 03:36 AM

(a) Apply row operations to the table

until it takes the form

, where

is the

identity matrix.

(b) Hence, or otherwise, give the inverse of the matrix

.

**math2009** · February 17th 2009, 04:15 AM

(a) $rref:\begin{bmatrix} -3&-2&1&0 \\ -3&1&0&1 \end{bmatrix}\begin{array}{c} \ \\ -R_1 \end{array}\longrightarrow\begin{bmatrix} -3&-2&1&0 \\ 0&3&-1&1 \end{bmatrix}\begin{array}{c} \div (-3) \\ \div 3 \end{array}\longrightarrow$ $\begin{bmatrix} 1&\frac{2}{3}&-\frac{1}{3}&0 \\ 0&1&-\frac{1}{3}&\frac{1}{3} \end{bmatrix}\begin{array}{c} -\frac{2}{3}R_2 \\ \ \end{array}\longrightarrow\begin{bmatrix} 1&0&-\frac{1}{9}&-\frac{2}{9} \\ 0&1&-\frac{1}{3}&\frac{1}{3} \end{bmatrix}$

(b) $A=\begin{bmatrix} a&b \\ c&d \end{bmatrix},\ A^{-1}=\frac{1}{ad-bc}\begin{bmatrix} d&-b \\ -c&a \end{bmatrix}=-\frac{1}{9}\begin{bmatrix} 1&2 \\ 3&-3 \end{bmatrix}$

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