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Chapter 2: Q32Q (page 93)

Find the inverse of matrices in Exercise 29-32, if they exist. Use the algorithm introduced in this section.

\(\left( {\begin{aligned}{*{20}{c}}{\bf{1}}&{ - {\bf{2}}}&{\bf{1}}\\{\bf{4}}&{ - {\bf{7}}}&{\bf{3}}\\{ - {\bf{2}}}&{\bf{6}}&{ - {\bf{4}}}\end{aligned}} \right)\)

Short Answer

Expert verified

The inverse of the matrix is not possible.

Step by step solution

01

Find the expression \(\left( {\begin{aligned}{*{20}{c}}A&I\end{aligned}} \right)\)

\(\left( {\begin{aligned}{*{20}{c}}A&I\end{aligned}} \right) = \left( {\begin{aligned}{*{20}{c}}1&{ - 2}&1\\4&{ - 7}&3\\{ - 2}&6&{ - 4}\end{aligned}\,\,\,\begin{aligned}{*{20}{c}}1&0&0\\0&1&0\\0&0&1\end{aligned}} \right)\)

02

Apply row operation to the matrix \(\left( {\begin{aligned}{*{20}{c}}A&I\end{aligned}} \right)\)

At row three, multiply row one by 2 and add it to row three, i.e., \({R_3} \to {R_3} + 2{R_1}\).

\(\left( {\begin{aligned}{*{20}{c}}A&I\end{aligned}} \right) = \left( {\begin{aligned}{*{20}{c}}1&{ - 2}&1\\4&{ - 7}&3\\0&2&{ - 2}\end{aligned}\,\,\,\begin{aligned}{*{20}{c}}1&0&0\\0&1&0\\2&0&1\end{aligned}} \right)\)

03

Apply row operation to the matrix \(\left( {\begin{aligned}{*{20}{c}}A&I\end{aligned}} \right)\)

At row two, multiply row one by 4 and subtract it from row two, i.e., \({R_2} \to {R_2} - 4{R_1}\).

\(\left( {\begin{aligned}{*{20}{c}}A&I\end{aligned}} \right) = \left( {\begin{aligned}{*{20}{c}}1&{ - 2}&1\\0&1&{ - 1}\\0&2&{ - 2}\end{aligned}\,\,\,\begin{aligned}{*{20}{c}}1&0&0\\{ - 4}&1&0\\2&0&1\end{aligned}} \right)\)

04

Apply row operation to the matrix \(\left( {\begin{aligned}{*{20}{c}}A&I\end{aligned}} \right)\)

At row three, multiply row two by 2 and subtract it from row three, i.e., \({R_3} \to {R_3} - 2{R_2}\).

\(\left( {\begin{aligned}{*{20}{c}}A&I\end{aligned}} \right) = \left( {\begin{aligned}{*{20}{c}}1&{ - 2}&1\\0&1&{ - 1}\\0&0&0\end{aligned}\,\,\,\begin{aligned}{*{20}{c}}1&0&0\\{ - 4}&1&0\\{10}&{ - 2}&1\end{aligned}} \right)\)

So, the inverse of the matrix is not possible.

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Most popular questions from this chapter

Explain why the columns of an \(n \times n\) matrix Aspan \({\mathbb{R}^{\bf{n}}}\) when

Ais invertible. (Hint:Review Theorem 4 in Section 1.4.)

Let \(A = \left[ {\begin{array}{*{20}{c}}B&{\bf{0}}\\{\bf{0}}&C\end{array}} \right]\), where \(B\) and \(C\) are square. Show that \(A\)is invertible if an only if both \(B\) and \(C\) are invertible.

Suppose Tand Ssatisfy the invertibility equations (1) and (2), where T is a linear transformation. Show directly that Sis a linear transformation. [Hint: Given u, v in \({\mathbb{R}^n}\), let \[{\mathop{\rm x}\nolimits} = S\left( {\mathop{\rm u}\nolimits} \right),{\mathop{\rm y}\nolimits} = S\left( {\mathop{\rm v}\nolimits} \right)\]. Then \(T\left( {\mathop{\rm x}\nolimits} \right) = {\mathop{\rm u}\nolimits} \), \[T\left( {\mathop{\rm y}\nolimits} \right) = {\mathop{\rm v}\nolimits} \]. Why? Apply Sto both sides of the equation \(T\left( {\mathop{\rm x}\nolimits} \right) + T\left( {\mathop{\rm y}\nolimits} \right) = T\left( {{\mathop{\rm x}\nolimits} + y} \right)\). Also, consider \(T\left( {cx} \right) = cT\left( x \right)\).]

Show that block upper triangular matrix \(A\) in Example 5is invertible if and only if both \({A_{{\bf{11}}}}\) and \({A_{{\bf{12}}}}\) are invertible. [Hint: If \({A_{{\bf{11}}}}\) and \({A_{{\bf{12}}}}\) are invertible, the formula for \({A^{ - {\bf{1}}}}\) given in Example 5 actually works as the inverse of \(A\).] This fact about \(A\) is an important part of several computer algorithims that estimates eigenvalues of matrices. Eigenvalues are discussed in chapter 5.

Prove Theorem 2(d). (Hint: The \(\left( {i,j} \right)\)- entry in \(\left( {rA} \right)B\) is \(\left( {r{a_{i1}}} \right){b_{1j}} + ... + \left( {r{a_{in}}} \right){b_{nj}}\).)
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