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Chapter 2: Q15Q (page 93)

Suppose A, B,and Care invertible \(n \times n\) matrices. Show that ABCis also invertible by producing a matrix Dsuch that \(\left( {ABC} \right)D = I\) and \(D\left( {ABC} \right) = I\).

Short Answer

Expert verified

It is proved that \(\left( {ABC} \right)D = I\) and \(D\left( {ABC} \right) = I\).

Step by step solution

01

Condition for an invertible matrix

Theorem 6 states that if Aand B are \(n \times n\) invertible matrices, the inverse of AB is the product of the inverses of Aand Bin the reverse order. That is, \({\left( {AB} \right)^{ - 1}} = {B^{ - 1}}{A^{ - 1}}\).

02

Show that \(\left( {ABC} \right)D = I\) and \(D\left( {ABC} \right) = I\)

Consider \(D = {C^{ - 1}}{B^{ - 1}}{A^{ - 1}}\). Then,

\(\begin{aligned}{c}\left( {ABC} \right){C^{ - 1}}{B^{ - 1}}{A^{ - 1}} = AB\left( {C{C^{ - 1}}} \right){B^{ - 1}}A\\ = ABI{B^{ - 1}}{A^{ - 1}}\\ = AI{A^{ - 1}}\\ = I\end{aligned}\)

And

\(\begin{aligned}{c}{C^{ - 1}}{B^{ - 1}}{A^{ - 1}}\left( {ABC} \right) = {C^{ - 1}}{B^{ - 1}}{A^{ - 1}}ABC\\ = {C^{ - 1}}{B^{ - 1}}IBC\\ = {C^{ - 1}}IC\\ = I\end{aligned}\)

Hence, it is proved that \(\left( {ABC} \right)D = I\) and \(D\left( {ABC} \right) = I\).

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Most popular questions from this chapter

Suppose the transfer function W(s) in Exercise 19 is invertible for some s. It can be showed that the inverse transfer function \(W{\left( s \right)^{ - {\bf{1}}}}\), which transforms outputs into inputs, is the Schur complement of \(A - BC - s{I_n}\) for the matrix below. Find the Sachur complement. See Exercise 15.

\(\left[ {\begin{array}{*{20}{c}}{A - BC - s{I_n}}&B\\{ - C}&{{I_m}}\end{array}} \right]\)

In Exercises 1鈥�9, assume that the matrices are partitioned conformably for block multiplication. Compute the products shown in Exercises 1鈥�4.

4. \[\left[ {\begin{array}{*{20}{c}}I&0\\{ - X}&I\end{array}} \right]\left[ {\begin{array}{*{20}{c}}A&B\\C&D\end{array}} \right]\]

In Exercises 1鈥�9, assume that the matrices are partitioned conformably for block multiplication. In Exercises 5鈥�8, find formulas for X, Y, and Zin terms of A, B, and C, and justify your calculations. In some cases, you may need to make assumptions about the size of a matrix in order to produce a formula. [Hint:Compute the product on the left, and set it equal to the right side.]

6. \[\left[ {\begin{array}{*{20}{c}}X&{\bf{0}}\\Y&Z\end{array}} \right]\left[ {\begin{array}{*{20}{c}}A&{\bf{0}}\\B&C\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}I&{\bf{0}}\\{\bf{0}}&I\end{array}} \right]\]

Generalize the idea of Exercise 21(a) [not 21(b)] by constructing a \(5 \times 5\) matrix \(M = \left[ {\begin{array}{*{20}{c}}A&0\\C&D\end{array}} \right]\) such that \({M^2} = I\). Make C a nonzero \(2 \times 3\) matrix. Show that your construction works.

Suppose Ais an \(m \times n\) matrix and there exist \(n \times m\) matrices C and D such that \(CA = {I_n}\) and \(AD = {I_m}\). Prove that \(m = n\) and \(C = D\). (Hint: Think about the product CAD.)
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