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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: Q2E (page 85)

If possible, compute the matrix products in Exercises 1 through 13, using paper and pencil.
2. $[\begin{matrix} 1 & - 1 \\ 0 & 2 \\ 2 & 1 \end{matrix}] [\begin{matrix} 3 & 2 \\ 1 & 0 \end{matrix}]$

Short Answer

Expert verified

The product of the given matrix is . $[\begin{matrix} 2 & 2 \\ 2 & 0 \\ 7 & 4 \end{matrix}]$

Step by step solution

01

Step1:Matrix multiplication

If A is matrix of order $n 脳 p$ and B is matrix of order $m 脳 q$ . Then the matrix multiplication of $A B$ Is defined only if. $p = m$

If Bis a $m 脳 q$ matrix and A $n 脳 p$ matrix, then the product BAis defined as the matrix of the linear transformation. $T (\overset{鈫�}{x}) = B (A \overset{鈫�}{x})$

02

Assuming the matrix

Let the given matrix

$A = [\begin{matrix} 1 & 鈭� 1 \\ 0 & 2 \\ 2 & 1 \end{matrix}], B = [\begin{matrix} 3 & 2 \\ 1 & 0 \end{matrix}]$

Order of Matrix $A$ is $3 脳 2$ and order of matrix $B$ is. $2 脳 2$

Since. $2 = 2$ Thus, the product is defined.

03

Multiplication of matrix

Now, find the product as follows:

$A B = [\begin{matrix} 3.1 鈭� 1.1 & 1.2 + (鈭� 1) .0 \\ 0.3 + 2.1 & 0.2 + 2.0 \\ 2.3 + 1.1 & 2.2 + 1.0 \end{matrix}] = [\begin{matrix} 2 & 2 \\ 2 & 0 \\ 7 & 4 \end{matrix}]$

Hence, the product of the given matrix is. $[\begin{matrix} 2 & 2 \\ 2 & 0 \\ 7 & 4 \end{matrix}]$

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Most popular questions from this chapter

If A is a $3 脳 4$ matrix and B is a $4 脳 5$ , then AB will be a $5 脳 3$ matrix.

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 ${鈩�}^{3}$ .

TRUE OR FALSE?

There exists a matrix A such that $A (\begin{matrix} \begin{matrix} 1 & 2 \\ 1 & 2 \end{matrix} \end{matrix}) = (\begin{matrix} 1 & 1 \\ 1 & 1 \end{matrix})$ .

Let $A = [\begin{matrix} 1 & 10 \\ - 3 & 12 \end{matrix}]$ in all parts of this problem.

(a) Find the scalar $位$ such that the matrix $A - {位濒}_{2}$ fails to be invertible. There are two solutions; choose one and use it in parts (b) and (c).

(b) For the $位$ you choose in part (a), find a non-zero vector $\overset{鈬赌}{x}$ such that

role="math" localid="1659697491583" $(A - {位濒}_{2}) \overset{鈬赌}{x} = \overset{鈬赌}{0}$

(c) Note that the equation $(A - {位濒}_{2}) \overset{鈬赌}{x} = \overset{鈬赌}{0}$ can be written as.

$A - 位 \overset{鈬赌}{x} = \overset{鈬赌}{0} o r A \overset{鈬赌}{x} = 位 \overset{鈬赌}{x}$ Check that the equation $A \overset{鈬赌}{x} = 位 \overset{鈬赌}{x}$ holds for your $位$ from part (a) and your $\overset{鈬赌}{x}$ from part (b).

If A is any transition matrix and B is any positive transition matrix, then AB must be a positive transition matrix.

See all solutions

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