Every square matrix with real-number elements has a number associated with it called its determinant. The determinant of a matrix can help us determine whether the matrix has an inverse. Let $A$ be a $n \times n$ matrix.

If $det A = 0,$ then $A$ is a singular matrix and has no inverse.
If $det A \neq = 0,$ then $A$ has an inverse matrix, written $A^{- 1},$ such that $A A^{- 1} = A^{- 1} A = I .$

Let's find the determinant of the given matrix.

Determinant

To determine whether the given $3 \times 3$ matrix has an inverse, we will calculate its determinant. To do so, we will start by copying the $first$ and $second$ columns and writing them on the right-hand side.

Now, we need to draw diagonals. Three diagonals will begin with the $upper - left$ $numbers$ and move down to the right. Another three will begin with the $upper - right$ $numbers$ and move down to the left.

Next, we multiply the numbers in each diagonal.

0 \cdot 4 \cdot 1 = 0 0 \cdot (- 2) \cdot 3 = 0 2 \cdot 1 \cdot (- 2) = - 4 3 \cdot 4 \cdot 2 = 24 - 2 \cdot (- 2) \cdot 0 = 0 1 \cdot 1 \cdot 0 = 0

Finally, we need to find the sum of the products in each set of diagonals and subtract the

second

sum

from the

first

sum .

(0 + 0 + (- 4)) - (24 + 0 + 0)

IdPropAdd

Identity Property of Addition

- 4 - 24

SubTerms

Subtract terms

- 28

Since the determinant of the matrix is not

0,

the matrix has an inverse.

Inverse

To find the inverse we will use a calculator. To create a matrix, we push $2 ND$ and $x^{- 1} .$ Then, using the right arrow, we navigate to the Edit tab and push $ENTER .$ The matrix is entered by defining the dimensions, $3 \times 3,$ and inputting the elements.

Now that we have entered the matrix we push $2 ND$ followed by $MODE$ to go back to the main screen. Then, we push $2 ND$ and $x^{- 1},$ select our matrix, and push $ENTER .$

Finally, we push $x^{- 1},$ followed by $ENTER .$ This will return us the inverse of our matrix!

We cannot properly see the elements of the inverse matrix, which are written as decimals. Let's express them as fractions. To do so, press $MODE,$ scroll down to the Answers option, and select Frac. Then, we go back to the main screen by pushing $2 ND$ and $MODE,$ and push $ENTER$ one last time.

Exercise

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