Question #323640

1.9.given A [111223400]\begin{bmatrix} 1 & 1& -1 \\ 2 & 2 & 3 \\ 4 & 0 &0 \\ \end{bmatrix} B=[111223400]\begin{bmatrix} 1 & 1 & -1 \\ 2 & 2 & 3 \\ 4 & 0 & 0 \\ \end{bmatrix}


1.9.1. find -A -1 +3BT

1.9.2. Find B-1+(AT+A-1)



1
Expert's answer
2022-04-05T16:23:29-0400

Remind that for 3×33\times3 matrix T=(a11a12a13a21a22a23a31a32a33)T=\left(\begin{array}{lll}a_{11}&a_{12}&a_{13}\\a_{21}&a_{22}&a_{23}\\a_{31}&a_{32}&a_{33}\end{array}\right) the inverse can be computed using the formula: T1=1TAdjTT^{-1}=\frac{1}{|T|}Adj\,\,T, where T|T| is the determinant of matrix TT. It has the form: T=a11a22a33+a21a13a32+a31a12a23a11a23a32a12a21a33a31a13a22|T|=a_{11}a_{22}a_{33}+a_{21}a_{13}a_{32}+a_{31}a_{12}a_{23}-a_{11}a_{23}a_{32}-a_{12}a_{21}a_{33}-a_{31}a_{13}a_{22} ; AdjT=(A11A12A13A21A22A23A31A32A33)Adj\,\,T=\left(\begin{array}{lll}A_{11}&A_{12}&A_{13}\\A_{21}&A_{22}&A_{23}\\A_{31}&A_{32}&A_{33}\end{array}\right) and A11=a22a23a32a33A_{11}=\left|\begin{array}{lll}a_{22}&a_{23}\\a_{32}&a_{33}\end{array}\right|, A12=a12a13a32a33A_{12}=-\left|\begin{array}{lll}a_{12}&a_{13}\\a_{32}&a_{33}\end{array}\right|, A13=a12a13a22a23A_{13}=\left|\begin{array}{lll}a_{12}&a_{13}\\a_{22}&a_{23}\end{array}\right|, A21=a21a23a31a33A_{21}=-\left|\begin{array}{lll}a_{21}&a_{23}\\a_{31}&a_{33}\end{array}\right|, A22=a11a13a31a33A_{22}=\left|\begin{array}{lll}a_{11}&a_{13}\\a_{31}&a_{33}\end{array}\right|, A23=a11a13a21a23A_{23}=-\left|\begin{array}{lll}a_{11}&a_{13}\\a_{21}&a_{23}\end{array}\right| , A31=a21a22a31a32A_{31}=\left|\begin{array}{lll}a_{21}&a_{22}\\a_{31}&a_{32}\end{array}\right|, A32=a11a12a31a32A_{32}=-\left|\begin{array}{lll}a_{11}&a_{12}\\a_{31}&a_{32}\end{array}\right|, A33=a11a12a21a22.A_{33}=\left|\begin{array}{lll}a_{11}&a_{12}\\a_{21}&a_{22}\end{array}\right|.

For the given matrices we have:

a). A1=(001435151425150)A^{-1}=\left(\begin{array}{lll}0&0&\frac{1}{4}\\\frac{3}{5}&\frac{1}{5}&-\frac{1}{4}\\-\frac{2}{5}&\frac{1}{5}&0\end{array}\right); B=(124120130)B^{\top}=\left(\begin{array}{lll}1&2&4\\1&2&0\\-1&3&0\end{array}\right);

A1+3B=(36474125295141354450)-A^{-1}+3B^{\top}=\left(\begin{array}{lll}3&6&\frac{47}{4}\\\frac{12}{5}&\frac{29}{5}&\frac{1}{4}\\-\frac{13}{5}&\frac{44}{5}&0\end{array}\right);

b). B1=A1B^{-1}=A^{-1} . We receive:

B1+(A+A1)=2A1+A=(129211512512951750)B^{-1}+(A^{\top}+A^{-1})=2A^{-1}+A^{\top}=\left(\begin{array}{lll}1&2&\frac{9}{2}\\\frac{11}{5}&\frac{12}{5}&-\frac{1}{2}\\-\frac{9}{5}&\frac{17}{5}&0\end{array}\right).


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