# For what value(s) of x does A^-1 exist.

• thatguythere
In summary, the question is asking for which values of x the matrix A-1 exists. The answer is any real number except for x = 3 or x = -3, as determined by the concept of invertibility and the determinant of the matrix A.
thatguythere
1. Homework Statement
Let A =
(x+1 2 )
(x+5 x+1)

For what value(s) of x, if any, does A-1 exist?

## The Attempt at a Solution

So let me see if I get the gist of this. An n x n matrix is invertible if and only if rank (A) = n. So unless there is a value of x which would make rank A < 2, then it is invertible. Since I can see no way to reduce both x+5 and x+1 to zero using one value, then A is invertible for all real numbers?

I don't know if you have learned the concept of determinants, but a matrix A is invertible if and only if its determinant is nonzero. The determinant for 2x2 matrices of the form
(a b)
(c d)
is given by det A = ad - bc

Computing the determinant and factoring the quadratic polynomial you'll get, will give you the answer.

Aha. So, det(A) = ((x+1)(x+1))-(2(x+5)) = x2-9 = (x+3)(x-3)
So A-1 exists for all values ≠ 3, -3.

Last edited:

## What does the term "A^-1" mean in this context?

The term "A^-1" refers to the inverse of matrix A. This is a matrix that, when multiplied by matrix A, results in the identity matrix. In other words, it "undoes" the effects of matrix A.

## Why is it important to know for what values of x the inverse of matrix A exists?

Knowing for what values of x the inverse of matrix A exists is important because it determines whether or not matrix A is invertible. If A^-1 exists, then matrix A is invertible and has a unique solution. However, if A^-1 does not exist, then matrix A is not invertible and does not have a unique solution.

## How can we determine for what values of x the inverse of matrix A exists?

The inverse of matrix A exists if and only if the determinant of A is not equal to 0. Therefore, to determine for what values of x the inverse of matrix A exists, we need to solve the equation det(A) ≠ 0 for x.

## What happens if we try to find the inverse of matrix A when it does not exist?

If A^-1 does not exist, then we cannot find the inverse of matrix A. This means that we cannot "undo" the effects of matrix A and find a unique solution. In this case, we may need to use other methods, such as row reduction or finding the pseudo-inverse, to solve for x.

## Can the inverse of a matrix exist for some values of x but not for others?

Yes, the inverse of a matrix can exist for some values of x but not for others. This is because the determinant of matrix A may be equal to 0 for some values of x, making A^-1 non-existent, but not equal to 0 for other values of x, making A^-1 existent.

### Similar threads

• Calculus and Beyond Homework Help
Replies
8
Views
918
• Calculus and Beyond Homework Help
Replies
22
Views
663
• Calculus and Beyond Homework Help
Replies
19
Views
509
• Calculus and Beyond Homework Help
Replies
1
Views
592
• Calculus and Beyond Homework Help
Replies
3
Views
759
• Calculus and Beyond Homework Help
Replies
2
Views
754
• Calculus and Beyond Homework Help
Replies
2
Views
576
• Calculus and Beyond Homework Help
Replies
4
Views
660
• Calculus and Beyond Homework Help
Replies
6
Views
1K
• Calculus and Beyond Homework Help
Replies
11
Views
230