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Chapter 9: Problem 55

What is the multiplicative identity matrix?

Short Answer

Expert verified
The multiplicative identity matrix is a special square matrix that leaves other matrices unchanged when they are multiplied by it. It's denoted by \(I\) and has 1s on the main diagonal and 0s elsewhere.

Step by step solution

01

Definition

The multiplicative identity matrix is a special type of square matrix. It is an array of numbers arranged in a square pattern, that when used in multiplication leaves the other matrix unchanged.
02

Identifying its Structure

The identity matrix, often denoted as \(I\), has 1s on the main diagonal and 0s in all other places (elements outside the main diagonal). This is its unique property and definition.
03

Presenting an Example

For instance, the 3x3 identity matrix is: \[I_3 = \begin{bmatrix} 1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & 1 \end{bmatrix}\] Any matrix multiplied by its corresponding identity matrix equals the matrix itself.

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Most popular questions from this chapter

Let $$ \begin{aligned} A &=\left[\begin{array}{cc} {1} & {0} \\ {0} & {1} \end{array}\right], \quad B=\left[\begin{array}{rr} {1} & {0} \\ {0} & {-1} \end{array}\right], \quad C=\left[\begin{array}{rr} {-1} & {0} \\ {0} & {1} \end{array}\right] \\ D &=\left[\begin{array}{rr} {-1} & {0} \\ {0} & {-1} \end{array}\right] \end{aligned} $$ Find the product of the difference between A and B and the sum of C and D.

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

Explaining the Concepts What is a matrix?

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

Solve: \(2 \cos ^{2} x+3 \sin x-3=0, \quad 0 \leq x<2 \pi\)
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