Math  /  Algebra

QuestionFind the inverse of [1111]\left[\begin{array}{cc}-1 & -1 \\ -1 & -1\end{array}\right]. If the inverse does not exist, select "undefined". Write each matrix element in simplest form.

Studdy Solution

STEP 1

What is this asking? We need to find the inverse of a 2x2 matrix, or explain why it doesn't have one! Watch out! Not all matrices have inverses!
We need to check for that *before* diving into calculations.

STEP 2

1. Calculate the determinant
2. Find the inverse (or explain why it's undefined)

STEP 3

Alright, let's **kick things off** by calculating the determinant!
The determinant of a 2x2 matrix [abcd]\begin{bmatrix} a & b \\ c & d \end{bmatrix} is given by adbcad - bc.
This little number tells us whether the matrix is invertible or not.

STEP 4

In our case, we have a=1a = -1, b=1b = -1, c=1c = -1, and d=1d = -1.
So, let's **plug these values** into our determinant formula: (1)(1)(1)(1)(-1) \cdot (-1) - (-1) \cdot (-1)

STEP 5

Now, let's **carefully evaluate** this expression: 11=01 - 1 = 0

STEP 6

Uh oh!
Our determinant is **zero**.
This tells us that the matrix *does not* have an inverse.

STEP 7

Since our determinant is **zero**, the matrix is **singular**, which means it doesn't have an inverse.
That's it!
We're done!

STEP 8

The inverse of the given matrix is **undefined**.

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