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Chapter 9: Problem 87

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. \((A+B)^{-1}=A^{-1}+B^{-1},\) assuming \(A, B,\) and \(A+B\) are invertible.

Short Answer

Expert verified
The statement is False. There's no corrected expression for the given equation on general grounds, unless additional conditions are specified.

Step by step solution

01

Understand the problem

The problem gives two operations on invertible matrices \(A\) and \(B\): The first operation sum of the matrices and then inversion of the sum i.e. \((A+B)^{-1}\); And the second operation, inverting the individual matrices first and then summing them i.e. \(A^{-1} + B^{-1}\). Here, we need to determine whether both operations yield the same result.
02

Evaluate both sides of the equation

If \(A,\) \(B,\) and \(A+B\) are invertible, \((A+B)^{-1}\) and \(A^{-1} + B^{-1}\) can be computed. However, it is widely known from the properties of matrix inversions that the inverse of a sum is not equals to the sum of the inverses. In simpler terms, \((A+B)^{-1}\) does not equal \(A^{-1} + B^{-1}\). Thus, the given statement can be stated to be false.
03

Provide the correct expression by recalling matrix inverse property

One of the commonly used properties of matrix inverse is that the inverse of the product of two matrices equals the product of their inverses in a reverse order, i.e., \((AB)^{-1}=B^{-1}A^{-1}\). But there is no similar property that equates the inverse of a sum of matrices to the sum of their inverses. Hence, there鈥檚 no corrected expression for the provided expression, unless additional conditions are specified.

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Most popular questions from this chapter

Write a system of linear equations in three or four variables to solve. Then use matrices to solve the system. Three foods have the following nutritional content per ounce. $$ \begin{array}{lccc} {} & {} & {\text { Protein }} & {\text { Vitamin } \mathrm{C}} \\ & {\text { Calories }} & {\text { (in grams) }} & {\text { (in milligrams) }} \\\ \hline \text { Food } A & {40} & {5} & {30} \\ {\text { Food } B} & {200} & {2} & {10} \\ {\text { Food } C} & {400} & {4} & {300} \end{array} $$ If a meal consisting of the three foods allows exactly 660 calories, 25 grams of protein, and 425 milligrams of vitamin \(\mathrm{C},\) how many ounces of each kind of food should be used?

Find (if possible) the following matrices: a. \(A B\) b. \(B A\) $$ A=\left[\begin{array}{ll} {4} & {2} \\ {6} & {1} \\ {3} & {5} \end{array}\right], \quad B=\left[\begin{array}{rrr} {2} & {3} & {4} \\ {-1} & {-2} & {0} \end{array}\right] $$

Solve: \(\quad 3^{2 x-8}=27 .\) (Section \(4.4,\) Example 1 )

Let $$ A=\left[\begin{array}{rr} {-3} & {-7} \\ {2} & {-9} \\ {5} & {0} \end{array}\right] \text { and } B=\left[\begin{array}{rr} {-5} & {-1} \\ {0} & {0} \\ {3} & {-4} \end{array}\right] $$ Solve each matrix equation for X. $$ 2 X+A=B $$

The figure shows the letter \(L\) in a rectangular coordinate system. The figure can be represented by the matrix $$ B=\left[\begin{array}{llllll} {0} & {3} & {3} & {1} & {1} & {0} \\ {0} & {0} & {1} & {1} & {5} & {5} \end{array}\right] $$ Each column in the matrix describes a point on the letter. The order of the columns shows the direction in which a pencil must move to draw the letter. The L is completed by connecting the last point in the matrix, (0,5), to the starting point, (0,0) . Use these ideas to solve Exercises 53-60. a. If \(A=\left[\begin{array}{rr}{-1} & {0} \\ {0} & {1}\end{array}\right],\) find \(A B\). b. Graph the object represented by matrix \(A B .\) What effect does the matrix multiplication have on the letter \(\mathrm{L}\) represented by matrix \(B ?\)
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