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Chapter 9: Problem 79

Determine whether each statement makes sense or does not make sense, and explain your reasoning. I'm working with two matrices that can be added but not multiplied.

Short Answer

Expert verified
The statement makes sense because while matrix addition requires similarity in dimension of matrices, matrix multiplication needs the number of columns in the first matrix to match the number of rows in the second matrix.

Step by step solution

01

Understanding Matrix Addition

Matrix addition is only possible when the two matrices have the same dimensions. That is, they must have the same number of rows and the same number of columns.
02

Understanding Matrix Multiplication

Matrix multiplication, unlike addition, does not require the two matrices to have the same dimensions. Instead, the number of columns in the first matrix must be equal to the number of rows in the second matrix.
03

Evaluating the Statement

If two matrices can be added, that means they have the same dimensions. But this doesn't necessarily mean they can be multiplied, because multiplication requires the number of columns in the first matrix to match the number of rows in the second. Hence, the statement makes sense.

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Most popular questions from this chapter

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 \(L\) represented by matrix \(B ?\)

Solve each system of equations using matrices. Use Gaussian elimination with back-substitution or Gauss-Jordan elimination. $$ \left\\{\begin{array}{r} {3 a-b-4 c=3} \\ {2 a-b+2 c=-8} \\ {a+2 b-3 c=9} \end{array}\right. $$

Explaining the Concepts Describe what is meant by the augmented matrix of a system of linear equations.

Solve each system of equations using matrices. Use Gaussian elimination with back-substitution or Gauss-Jordan elimination. $$ \left\\{\begin{aligned} 3 a+b-c &=0 \\ 2 a+3 b-5 c &=1 \\ a-2 b+3 c &=-4 \end{aligned}\right. $$

Perform the indicated matrix operations given that \(A, B,\) and \(C\) are defined as follows. If an operation is not defined, state the reason. $$ A=\left[\begin{array}{rr} {4} & {0} \\ {-3} & {5} \\ {0} & {1} \end{array}\right] \quad B=\left[\begin{array}{rr} {5} & {1} \\ {-2} & {-2} \end{array}\right] \quad C=\left[\begin{array}{rr} {1} & {-1} \\ {-1} & {1} \end{array}\right] $$ $$ 5 C-2 B $$
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