Identifying Matrices With Inverses: A Guide

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Hey guys! Let's dive into the world of matrices and figure out which ones have something special – an inverse. This is super important in linear algebra, and understanding it will open up a lot of possibilities. So, the question is: Assinale a alternativa que apresenta uma matriz que possui inversa. (Choose the alternative that presents a matrix with an inverse). We'll break down what an inverse is, how to spot one, and then go through the options you gave to find the right answer. Ready to get started?

Understanding Matrix Inverses

Alright, first things first: what is a matrix inverse? Think of it like this: in the regular number world, if you have a number, like 5, its inverse is 1/5 (because 5 * 1/5 = 1). The same concept applies to matrices. If you have a matrix A, its inverse (denoted as A⁻¹) is another matrix that, when multiplied by A, gives you the identity matrix. The identity matrix is like the number 1 in the matrix world – it doesn't change anything when you multiply by it. For a 2x2 matrix, the identity matrix looks like this: [[1, 0], [0, 1]]. For a 3x3 matrix, it's [[1, 0, 0], [0, 1, 0], [0, 0, 1]], and so on.

So, in short: A * A⁻¹ = I (where I is the identity matrix).

Now, here's the catch: not all matrices have inverses. Matrices that do have inverses are called invertible or non-singular matrices. Matrices that don't have inverses are called singular or non-invertible matrices. The existence of an inverse depends on a few things, but the most important one is the determinant.

The determinant is a special number associated with a square matrix. If the determinant of a matrix is not zero, then the matrix has an inverse. If the determinant is zero, then the matrix does not have an inverse (it's singular).

Let's keep it casual. Think of the determinant as a gatekeeper. If the determinant is non-zero, the matrix is allowed to have an inverse. If the determinant is zero, the gate is closed, and no inverse is allowed. Calculating the determinant is different for 2x2, 3x3, and larger matrices, but the principle is the same.

This is the core concept of this article, so understanding the determinant and its relationship to the inverse is crucial for answering the question accurately. We'll use this knowledge to evaluate the matrices provided in the question.

Checking the Options: Finding the Matrix with an Inverse

Okay, now that we know what we're looking for, let's examine the options and see which matrix has an inverse. We'll need to calculate the determinant of each matrix and determine if it's non-zero. Let's get to it!

Option A: 2x2 Matrix Analysis

The first option is a 2x2 matrix: A = [[2, 1], [4, 2]].

To find the determinant of a 2x2 matrix [[a, b], [c, d]], we use the formula: determinant = (a * d) - (b * c).

For our matrix A, this means: determinant = (2 * 2) - (1 * 4) = 4 - 4 = 0.

Since the determinant is 0, this matrix does not have an inverse. So, option A is out.

Option B: Another 2x2 Matrix Test

Next up, we have another 2x2 matrix: A = [[2, 1], [1, 1]].

Using the same determinant formula: determinant = (2 * 1) - (1 * 1) = 2 - 1 = 1.

Since the determinant is 1 (which is not zero), this matrix does have an inverse. Ding ding ding! We may have found our answer. However, we'll confirm by checking the remaining options.

Option C: Examining a 3x2 Matrix

Here, we are given a 3x2 matrix: A = [[1, 2], [3, 4], [6, 0]].

Important note: Only square matrices (matrices with the same number of rows and columns) can have inverses. Since this matrix is 3x2 (3 rows, 2 columns), it is not a square matrix. Therefore, it cannot have an inverse. So we can rule out option C immediately.

Option D: Analyzing a 3x3 Matrix

Now, let's look at this 3x3 matrix: A = [[1, 2, 1], [0, 0, 0], [1, 1, 4]].

To find the determinant of a 3x3 matrix, you can use several methods (cofactor expansion, for example). However, we can spot a shortcut here. Notice that the second row consists entirely of zeros. This means the determinant will be zero. Here's why: if you expand the determinant along the second row (using cofactor expansion), each term will be multiplied by zero, resulting in a zero determinant. Therefore, this matrix doesn't have an inverse.

Option E: 3x3 Matrix Check

The final option is another 3x3 matrix: A = [[1, 2, 0], [2, -1, 0], [1, 3, 0]].

Again, we can calculate the determinant. Notice that the third column consists of all zeros. This also will result in a zero determinant. To see this you could perform a cofactor expansion along the third column. Each value in the third column will be multiplied by its corresponding cofactor, and since the values are all zero, the result will be zero. Thus, this matrix is also not invertible.

Conclusion: The Winning Matrix!

After analyzing all the options, we found that only one matrix has an inverse. Option B: A = [[2, 1], [1, 1]] is the correct answer because its determinant is 1 (not zero), making it an invertible matrix. The other options either had a determinant of zero or were not square matrices and, therefore, did not possess an inverse. Awesome job, guys! You now know how to identify matrices that have inverses!

Additional Tips and Tricks

Here are some extra things to keep in mind when dealing with matrix inverses:

  • Determinant is Key: The determinant is always your first check. If it's zero, you know immediately there's no inverse. This will save you time.
  • Square Matrices Only: Only square matrices can have inverses. If it's not square, no need to calculate the determinant.
  • Recognize Patterns: Look for patterns in the matrix that might make the determinant easy to determine (like a row or column of zeros, or a row/column that is a multiple of another).
  • Calculating the Inverse (Optional): If you do find a matrix with an inverse, you can calculate the inverse using formulas (for 2x2 matrices) or more complex methods (for larger matrices). This usually involves finding the determinant, calculating the matrix of minors, cofactors, and the adjugate matrix.

Keep practicing, and you'll get really good at this. Understanding matrix inverses is not just about passing a test, it's about building a solid foundation in linear algebra. You'll use this knowledge in many other areas of mathematics, computer science, physics, and engineering. Good luck!