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Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 2: Q59E (page 86)

In Exercises 55 through 64, find all matricesX that satisfy the given matrix equation.
$X [\begin{matrix} 2 & 1 \\ 4 & 2 \end{matrix}] = l_{2}$

Short Answer

Expert verified

There is no such matrix X exists such that $X [\begin{matrix} 2 & 1 \\ 4 & 2 \end{matrix}] = l_{2}$ .

Step by step solution

01

Assuming the matrix

Since, the product of matrix gives $2 脳 2$ matrix then the order of matrix X should be $2 脳 2$

Let the matrix be

X = [\begin{matrix} a & c \\ b & d \end{matrix}]

.

We know the matrix $I_{2} = [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}]$

Then the equation will become

$[\begin{matrix} a & c \\ b & d \end{matrix}] [\begin{matrix} 2 & 1 \\ 4 & 2 \end{matrix}] = [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}]$

02

Solving the Matrix

To solve the given matrix equation, first multiply the left-hand side matrix.

$[\begin{matrix} 2 a + 4 c & a + 2 c \\ 2 b + 4 d & b + 2 d \end{matrix}] = [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}]$

Now equating the left-hand and right-hand matrix.

role="math" $2 a + 4 c = 1 a + 2 c = 0 2 b + 4 d = 0 b + 2 d = 1$

On solving the above equa3tions, we will get $0 = 1$ which is not possible for any value.

Hence, there is no such matrix X exists such that $X [\begin{matrix} 2 & 1 \\ 4 & 2 \end{matrix}] = I_{2}$

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Most popular questions from this chapter

Question:

If Ais any 3x3 transition matrix (see Definition 2.1.4), find the matrix product $[1 1 1] A$ .
For a fixed n , let $\overset{鈫�}{e}$ be the row vector $\overset{鈫�}{e} = \frac{[1, 1, . ., 1]}{n 1' s}$ . Show that an nxn matrix A with nonnegative entries is a transition matrixif (and only if) $\overset{鈫�}{e} A = \overset{鈫�}{e}$ .

Question: TRUE OR FALSE?

If T is any linear transformation from ${鈩�}^{3} t o {鈩�}^{3^{}}$ , then $T (\overset{鈫�}{v} 脳 \overset{鈫�}{w}) = T (\overset{鈫�}{v}) 脳 T (\overset{鈫�}{w})$ for all vectors $\overset{鈫�}{v}$ and $\overset{鈫�}{w}$ in ${鈩�}^{3}$ .

If A is any invertible $n 脳 n$ matrix, then A commutes with $A^{- 1}$ .

Matrixis invertible for all real numbers k.

If $A^{17} = l^{2}$ then the matrix A must be $l^{2}$ .

See all solutions

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