Chapter 11: Q. 17 (page 793)

In Problems 17–19, find the inverse, if there is one, of each matrix. If there is not an inverse, say that the matrix is singular.
$[\begin{matrix} 4 & 6 \\ 1 & 3 \end{matrix}]$

Short Answer

Expert verified

The inverse of the given matrix is $[\begin{matrix} \frac{1}{2} & - 1 \\ - \frac{1}{6} & \frac{2}{3} \end{matrix}]$

Step by step solution

Step 1. Given information

The given matrix is $[\begin{matrix} 4 & 6 \\ 1 & 3 \end{matrix}]$

Step 2. Find the inverse of the matrix.

$[A ∣ I_{2}] = [\begin{array}{llll} 4 & 6 & 1 & 0 \\ 1 & 3 & 0 & 1 \end{array}]$

Transform the matrix $[A ∣ I_{2}]$ into reduced row echelon form.

Perform the operations $R_{1} = \frac{1}{4} R_{1}$ and then localid="1646987914703" $R_{2} = R_{2} - R_{1}$ ,

$[\begin{array}{llll} 4 & 6 & 1 & 0 \\ 1 & 3 & 0 & 1 \end{array}] \to [\begin{array}{cccc} 1 & \frac{3}{2} & \frac{1}{4} & 0 \\ 1 & 3 & 0 & 1 \end{array}] \to [\begin{array}{cccc} 1 & \frac{3}{2} & \frac{1}{4} & 0 \\ 0 & \frac{3}{2} & - \frac{1}{4} & 1 \end{array}]$

Now, perform the operations $R_{1} = R_{1} - R_{2}$ and then $R_{2} = \frac{2}{3} R_{2}$ ,

$[\begin{array}{cccc} 1 & \frac{3}{2} & \frac{1}{4} & 0 \\ 0 & \frac{3}{2} & - \frac{1}{4} & 1 \end{array}] \to [\begin{array}{cccc} 1 & 0 & \frac{1}{2} & - 1 \\ 0 & \frac{3}{2} & - \frac{1}{4} & 1 \end{array}] \to [\begin{array}{ccrr} 1 & 0 & \frac{1}{2} & - 1 \\ 0 & 1 & - \frac{1}{6} & \frac{2}{3} \end{array}]$

Now, we can see that the identity matrix is on the left of the vertical bar.

Then the matrix on the right of the vertical bar is the inverse of $A$ .

Therefore, the inverse of the given matrix is $[\begin{matrix} \frac{1}{2} & - 1 \\ - \frac{1}{6} & \frac{2}{3} \end{matrix}]$

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In Problems 17–19, find the inverse, if there is one, of each matrix. If there is not an inverse, say that the matrix is singular.
$[\begin{matrix} 4 & 6 \\ 1 & 3 \end{matrix}]$

Short Answer

Step by step solution

Step 1. Given information

Step 2. Find the inverse of the matrix.

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In Problems 17–19, find the inverse, if there is one, of each matrix. If there is not an inverse, say that the matrix is singular.4613

Short Answer

Step by step solution

Step 1. Given information

Step 2. Find the inverse of the matrix.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Geometry

Discrete Mathematics

Calculus

Mechanics Maths

Logic and Functions

Study anywhere. Anytime. Across all devices.

In Problems 17–19, find the inverse, if there is one, of each matrix. If there is not an inverse, say that the matrix is singular.
$[\begin{matrix} 4 & 6 \\ 1 & 3 \end{matrix}]$