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Chapter 9: Problem 60

Determine whether each statement makes sense or does not make sense, and explain your reasoning. When I use matrices to solve linear systems, the only arithmetic involves multiplication or a combination of multiplication and addition.

Short Answer

Expert verified
The statement partially makes sense because while multiplication and a combination of multiplication and addition are involved in solving linear systems using matrices, subtraction and indirect division (multiplication by the reciprocal) are also necessary.

Step by step solution

01

Understand the statement

The statement is suggesting that the process of solving linear systems using matrices only involves two types of arithmetic: multiplication, or a combination of multiplication and addition.
02

Reflect on the process of solving linear systems using matrices

When we solve linear systems using matrices, we use row operations. Those operations are: (1) Swapping two rows (2) Multiplying a row by a non-zero constant (3) Adding a multiple of one row to another row. Clearly, alongside multiplication and addition, subtraction (when we subtract one row from another) is also a common operation. Furthermore, division is, indirectly, involved when we multiply a row by a non-zero constant (which is equivalent to multiplying by the reciprocal aka division).
03

Evaluate the statement

With the process of solving linear systems using matrices in mind, it is evident that the statement does not make complete sense. Yes, multiplication and addition are indeed involved, but solving matrices would be incomplete without subtraction and division.

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Most popular questions from this chapter

Find the following matrices: a. \(A+B\) b. \(A-B\) c. \(-4 A\) d. \(3 A+2 B\) $$ A=\left[\begin{array}{ll} {1} & {3} \\ {3} & {4} \\ {5} & {6} \end{array}\right], \quad B=\left[\begin{array}{rr} {2} & {-1} \\ {3} & {-2} \\ {0} & {1} \end{array}\right] $$

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The row operation \(k R_{i}+R_{j}\) indicates that it is the elements in row \(i\) that change.

Find (if possible) the following matrices: a. \(A B\) b. \(B A\) $$ A=\left[\begin{array}{l} {-1} \\ {-2} \\ {-3} \end{array}\right], \quad B=\left[\begin{array}{lll} {1} & {2} & {3} \end{array}\right] $$

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. Use matrix operations to move the L 2 units to the left and 3 units down. Then graph the letter and its transformation in a rectangular coordinate system.

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