Chapter 4.3, Problem 4.4E

(a)

Interpretation Introduction

Interpretation: The given multiplication of matrix needs to be performed.

  $\left[\begin{array}{ccc}5& 1& 3\\ 4& 2& 3\\ 1& 2& 3\end{array}\right]\times \left[\begin{array}{ccc}2& 1& 1\\ 1& 2& 3\\ 5& 4& 3\end{array}\right]$

Concept Introduction: A matrix is represented as an ordered array of numbers as follows:

  $\left[\begin{array}{cc}3& 7\\ 2& 1\end{array}\right]\text{ or, }\left[\begin{array}{ccc}2& 0& 1\end{array}\right]$

In order to multiply two matrices, in first matrix the number in vertical columns should be equal to the number of horizontal rows in the second matrix. To perform the multiplication, the product of each row in the first matrix needs to be added term by term on the each column of the second matrix. The resulting number after addition is placed in the product matrix with row obtained from the row of the first matrix and column obtained from the column of the second matrix.

The mathematical representation is as follows:

  $C_{ij}={\displaystyle \sum A_{ik}}\times B_{kj}$

Here, $C_{ij}$ is product matrix with i row and j column.

  $A_{ik}$ is initial matrix A with i row and k column.

  $B_{kj}$ is initial matrix B with k row and j column.

(b)

Interpretation Introduction

Interpretation: The given multiplication of matrix needs to be performed.

  $\left[\begin{array}{ccc}1& -1& -2\\ 0& 1& -1\\ 1& 0& 0\end{array}\right]\times \left[\begin{array}{c}2\\ 1\\ 3\end{array}\right]$

Concept Introduction: A matrix is represented as an ordered array of numbers as follows:

  $\left[\begin{array}{cc}3& 7\\ 2& 1\end{array}\right]\text{ or, }\left[\begin{array}{ccc}2& 0& 1\end{array}\right]$

In order to multiply two matrices, in first matrix the number in vertical columns should be equal to the number of horizontal rows in the second matrix. To perform the multiplication, the product of each row in the first matrix needs to be added term by term on the each column of the second matrix. The resulting number after addition is placed in the product matrix with row obtained from the row of the first matrix and column obtained from the column of the second matrix.

The mathematical representation is as follows:

  $C_{ij}={\displaystyle \sum A_{ik}}\times B_{kj}$

Here, $C_{ij}$ is product matrix with i row and j column.

  $A_{ik}$ is initial matrix A with i row and k column.

  $B_{kj}$ is initial matrix B with k row and j column.

(c)

Interpretation Introduction

Interpretation: The given multiplication of matrix needs to be performed.

  $\left[\begin{array}{ccc}1& 2& 3\end{array}\right]\times \left[\begin{array}{ccc}1& -1& -2\\ 2& 1& -1\\ 3& 2& 1\end{array}\right]$

Concept Introduction: A matrix is represented as an ordered array of numbers as follows:

  $\left[\begin{array}{cc}3& 7\\ 2& 1\end{array}\right]\text{ or, }\left[\begin{array}{ccc}2& 0& 1\end{array}\right]$

In order to multiply two matrices, in first matrix the number in vertical columns should be equal to the number of horizontal rows in the second matrix. To perform the multiplication, the product of each row in the first matrix needs to be added term by term on the each column of the second matrix. The resulting number after addition is placed in the product matrix with row obtained from the row of the first matrix and column obtained from the column of the second matrix.

The mathematical representation is as follows:

  $C_{ij}={\displaystyle \sum A_{ik}}\times B_{kj}$

Here, $C_{ij}$ is product matrix with i row and j column.

  $A_{ik}$ is initial matrix A with i row and k column.

  $B_{kj}$ is initial matrix B with k row and j column.