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Chapter 2: Q17Q (page 93)

Solve the equation \(AB = BC\) for A, assuming that A, B, and C are square and Bis invertible.

Short Answer

Expert verified

\(A = BC{B^{ - 1}}\)

Step by step solution

01

Condition for an invertible matrix

Theorem 5states that Ais an invertible \(n \times n\) matrix, then for each b in \({\mathbb{R}^n}\), the equation \(Ax = b\) has a unique solution \(x = {A^{ - 1}}b\).

02

Solve the equation AB = BC for A

Multiply both sides of the equation \(AB = BC\) by \({B^{ - 1}}\):

\(\begin{aligned}{c}AB{B^{ - 1}} = BC{B^{ - 1}}\\AI = BC{B^{ - 1}}\\A = BC{B^{ - 1}}\end{aligned}\)

Thus, \(A = BC{B^{ - 1}}\).

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Most popular questions from this chapter

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

1. \(\left[ {\begin{array}{*{20}{c}}I&{\bf{0}}\\E&I\end{array}} \right]\left[ {\begin{array}{*{20}{c}}A&B\\C&D\end{array}} \right]\)

1. Find the inverse of the matrix \(\left( {\begin{aligned}{*{20}{c}}{\bf{8}}&{\bf{6}}\\{\bf{5}}&{\bf{4}}\end{aligned}} \right)\).

Exercises 21 and 22 concern the way in which color is specified for display in computer graphics. A color on a computer screen is encoded by three numbers (R, G, B) that list the amount of energy an electron gun must transmit to red, green, and blue phosphor dots on the computer screen. (A fourth number specifies the luminance or intensity of the color.)

21. The actual color a viewer sees on a screen is influenced by the specific type and amount of phosphors on the screen. So, each computer screen manufacturer must convert between the (R, G, B) data and an international CIE standard for color, which uses three primary colors, called X, Y, and Z. A typical conversion for short-persistence phosphors is

\(\left[ {\begin{array}{*{20}{c}}{.61}&{.29}&{.150}\\{.35}&{.59}&{.063}\\{.04}&{.12}&{.787}\end{array}} \right]\left[ {\begin{array}{*{20}{c}}R\\G\\B\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}X\\Y\\Z\end{array}} \right]\)

A computer program will send a stream of color information to the screen, using standard CIE data (X, Y, Z). Find the equation that converts these data to the (R, G, B) data needed for the screen鈥檚 electron gun.

24. (QR Factorization) Suppose \[A = QR\], where Qand R are \[n \times n\], Ris invertible and upper triangular, and Q has the property that \[{Q^T}{\bf{Q}} = I\]. Show that for each b in \[{\mathbb{R}^n}\], the equation \[Ax = b\] has a unique solution. What computations with Q and R will produce the solution?

Let T be a linear transformation that maps \({\mathbb{R}^n}\) onto \({\mathbb{R}^n}\). Is \({T^{ - 1}}\) also one-to-one?

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