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Chapter 6: Problem 47

What is the multiplicative identity matrix?

Short Answer

Expert verified
The multiplicative identity matrix is a square matrix with ones on its diagonal and zeros everywhere else. When it is multiplied by any other matrix, the result is still the original matrix.

Step by step solution

01

Definition of an Identity Matrix

The multiplicative identity matrix, often simply called the identity matrix, is a square matrix (same number of rows as columns), denoted as I, with ones on the diagonal and zeros everywhere else. For a 2x2 matrix, it is: \[ \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix} \]
02

Properties of an Identity Matrix

One of the important properties of an identity matrix is that when it is multiplied by any other matrix, the original matrix is unchanged. It's symbolized as: If A is any m脳n matrix, then IA = A and AI = A.
03

Example of Multiplying with an Identity Matrix

For instance, if we have a matrix A such as: \[ \begin{bmatrix} a & b \ c & d \end{bmatrix} \] And multiply it with the 2x2 identity matrix I, we will still get the same matrix A. Like so: \[ \begin{bmatrix} a & b \ c & d \end{bmatrix} * \begin{bmatrix} 1 & 0 \ 0 & 1 \end{bmatrix} = \begin{bmatrix} a & b \ c & d \end{bmatrix} \]

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Most popular questions from this chapter

Consider the system $$ \begin{array}{l}a_{1} x+b_{1} y=c_{1} \\\a_{2} x+b_{2} y=c_{2}\end{array} $$ Use Cramer's rule to prove that if the first equation of the system is replaced by the sum of the two equations, the resulting system has the same solution as the original system.

Use Cramer's rule to solve each system or to determine that the system is inconsistent or contains dependent equations. $$ \begin{aligned}&2 x=3 y+2\\\&5 x=51-4 y\end{aligned} $$

In Exercises \(9-16,\) find: a. \(A+B\) b. \(A-B\) c. \(-4 A\) d. \(3 A+2 B\) $$ A=\left[\begin{array}{rrr} 2 & -10 & -2 \\ 14 & 12 & 10 \\ 4 & -2 & 2 \end{array}\right], \quad B=\left[\begin{array}{rrr} 6 & 10 & -2 \\ 0 & -12 & -4 \\ -5 & 2 & -2 \end{array}\right] $$

In Exercises \(17-26,\) let $$ A=\left[\begin{array}{rr} -3 & -7 \\ 2 & -9 \\ 5 & 0 \end{array}\right] \text { and } B=\left[\begin{array}{rr} -5 & -1 \\ 0 & 0 \\ 3 & -4 \end{array}\right] $$ Solve each matrix equation for \(X\). $$ X-A=B $$

Without going into too much detail, describe how to solve a linear system in three variables using Cramer's rule.
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