Find AB if ________________ ________________

Chapter 2, Problem 3E

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QUESTION:

Find AB if

Find \(\mathbf{A B}\) if

a) \(\mathbf{A}=\left[\begin{array}{ll}2 & 1 \\ 3 & 2\end{array}\right], \mathbf{B}=\left[\begin{array}{ll}0 & 4 \\ 1 & 3\end{array}\right]\).

b) \(\mathbf{A}=\left[\begin{array}{rr}1 & -1 \\ 0 & 1 \\ 2 & 3\end{array}\right], \mathbf{B}=\left[\begin{array}{rrr}3 & -2 & -1 \\ 1 & 0 & 2\end{array}\right]\).

c) \(\mathbf{A}=\left[\begin{array}{rr}4 & -3 \\ 3 & -1 \\ 0 & -2 \\ -1 & 5\end{array}\right], \mathbf{B}=\left[\begin{array}{rrrr}-1 & 3 & 2 & -2 \\ 0 & -1 & 4 & -3\end{array}\right]\).

<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>)</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mi>A</mi><mo>=</mo><mo>&#xA0;</mo><mfenced open="[" close="]"><mtable><mtr><mtd><mn>2</mn></mtd><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mn>3</mn></mtd><mtd><mn>2</mn></mtd></mtr></mtable></mfenced><mo>&#xA0;</mo><mo>,</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mi>B</mi><mo>&#xA0;</mo><mo>=</mo><mo>&#xA0;</mo><mfenced open="[" close="]"><mtable><mtr><mtd><mn>0</mn></mtd><mtd><mn>4</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd><mtd><mn>3</mn></mtd></mtr></mtable></mfenced><mo>&#xA0;</mo><mo>.</mo><mspace linebreak="newline"/><mi>b</mi><mo>)</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mi>A</mi><mo>&#xA0;</mo><mo>=</mo><mfenced open="[" close="]"><mtable><mtr><mtd><mn>1</mn></mtd><mtd><mo>-</mo><mn>1</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mn>2</mn></mtd><mtd><mn>3</mn></mtd></mtr></mtable></mfenced><mo>&#xA0;</mo><mo>,</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mi>B</mi><mo>&#xA0;</mo><mo>=</mo><mo>&#xA0;</mo><mfenced open="[" close="]"><mtable><mtr><mtd><mn>3</mn></mtd><mtd><mo>-</mo><mn>2</mn></mtd><mtd><mo>-</mo><mn>1</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>2</mn></mtd></mtr></mtable></mfenced><mo>&#xA0;</mo><mo>.</mo><mspace linebreak="newline"/><mi>c</mi><mo>)</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mi>A</mi><mo>&#xA0;</mo><mo>=</mo><mfenced open="[" close="]"><mtable><mtr><mtd><mn>4</mn></mtd><mtd><mo>-</mo><mn>3</mn></mtd></mtr><mtr><mtd><mn>3</mn></mtd><mtd><mo>-</mo><mn>1</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mo>-</mo><mn>2</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>1</mn></mtd><mtd><mn>5</mn></mtd></mtr></mtable></mfenced><mo>&#xA0;</mo><mo>,</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mi>B</mi><mo>&#xA0;</mo><mo>=</mo><mo>&#xA0;</mo><mfenced open="[" close="]"><mtable><mtr><mtd><mo>-</mo><mn>1</mn></mtd><mtd><mn>3</mn></mtd><mtd><mn>2</mn></mtd><mtd><mo>-</mo><mn>2</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mo>-</mo><mn>1</mn></mtd><mtd><mn>4</mn></mtd><mtd><mo>-</mo><mn>3</mn></mtd></mtr></mtable></mfenced><mo>&#xA0;</mo><mo>.</mo></math>

Equation Transcription:

[], []

[] []

[] []

Text Transcription:

AB

A=[ 3   2  2   1 ], B=[ 1  3 0   4 ]

A=[ 2   3  0  1 1    -1] B=[1  0    2 3   -2    -1]  

A=[-1   5  0   -2   3  -1  4 -3] B=[1  -1  4    -3 1  3   2  -2]

Questions & Answers

QUESTION:

Find AB if

Find \(\mathbf{A B}\) if

a) \(\mathbf{A}=\left[\begin{array}{ll}2 & 1 \\ 3 & 2\end{array}\right], \mathbf{B}=\left[\begin{array}{ll}0 & 4 \\ 1 & 3\end{array}\right]\).

b) \(\mathbf{A}=\left[\begin{array}{rr}1 & -1 \\ 0 & 1 \\ 2 & 3\end{array}\right], \mathbf{B}=\left[\begin{array}{rrr}3 & -2 & -1 \\ 1 & 0 & 2\end{array}\right]\).

c) \(\mathbf{A}=\left[\begin{array}{rr}4 & -3 \\ 3 & -1 \\ 0 & -2 \\ -1 & 5\end{array}\right], \mathbf{B}=\left[\begin{array}{rrrr}-1 & 3 & 2 & -2 \\ 0 & -1 & 4 & -3\end{array}\right]\).

<math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>)</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mi>A</mi><mo>=</mo><mo>&#xA0;</mo><mfenced open="[" close="]"><mtable><mtr><mtd><mn>2</mn></mtd><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mn>3</mn></mtd><mtd><mn>2</mn></mtd></mtr></mtable></mfenced><mo>&#xA0;</mo><mo>,</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mi>B</mi><mo>&#xA0;</mo><mo>=</mo><mo>&#xA0;</mo><mfenced open="[" close="]"><mtable><mtr><mtd><mn>0</mn></mtd><mtd><mn>4</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd><mtd><mn>3</mn></mtd></mtr></mtable></mfenced><mo>&#xA0;</mo><mo>.</mo><mspace linebreak="newline"/><mi>b</mi><mo>)</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mi>A</mi><mo>&#xA0;</mo><mo>=</mo><mfenced open="[" close="]"><mtable><mtr><mtd><mn>1</mn></mtd><mtd><mo>-</mo><mn>1</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mn>2</mn></mtd><mtd><mn>3</mn></mtd></mtr></mtable></mfenced><mo>&#xA0;</mo><mo>,</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mi>B</mi><mo>&#xA0;</mo><mo>=</mo><mo>&#xA0;</mo><mfenced open="[" close="]"><mtable><mtr><mtd><mn>3</mn></mtd><mtd><mo>-</mo><mn>2</mn></mtd><mtd><mo>-</mo><mn>1</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd><mtd><mn>0</mn></mtd><mtd><mn>2</mn></mtd></mtr></mtable></mfenced><mo>&#xA0;</mo><mo>.</mo><mspace linebreak="newline"/><mi>c</mi><mo>)</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mi>A</mi><mo>&#xA0;</mo><mo>=</mo><mfenced open="[" close="]"><mtable><mtr><mtd><mn>4</mn></mtd><mtd><mo>-</mo><mn>3</mn></mtd></mtr><mtr><mtd><mn>3</mn></mtd><mtd><mo>-</mo><mn>1</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mo>-</mo><mn>2</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>1</mn></mtd><mtd><mn>5</mn></mtd></mtr></mtable></mfenced><mo>&#xA0;</mo><mo>,</mo><mo>&#xA0;</mo><mo>&#xA0;</mo><mi>B</mi><mo>&#xA0;</mo><mo>=</mo><mo>&#xA0;</mo><mfenced open="[" close="]"><mtable><mtr><mtd><mo>-</mo><mn>1</mn></mtd><mtd><mn>3</mn></mtd><mtd><mn>2</mn></mtd><mtd><mo>-</mo><mn>2</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd><mtd><mo>-</mo><mn>1</mn></mtd><mtd><mn>4</mn></mtd><mtd><mo>-</mo><mn>3</mn></mtd></mtr></mtable></mfenced><mo>&#xA0;</mo><mo>.</mo></math>

Equation Transcription:

[], []

[] []

[] []

Text Transcription:

AB

A=[ 3   2  2   1 ], B=[ 1  3 0   4 ]

A=[ 2   3  0  1 1    -1] B=[1  0    2 3   -2    -1]  

A=[-1   5  0   -2   3  -1  4 -3] B=[1  -1  4    -3 1  3   2  -2]

ANSWER:

Solution:

Step 1:

In this problem we have to find AB matrix for these matrices.

Matrix Multiplication :  if A be any matrix of order and B is an matrix , then their matrix product AB is an  matrix , in which the m entries of row of A are multiplied with the m entries down a columns of B after then we summed to produce an entry of AB.

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