Matrix Invertibility Find The Value Of X For No Inverse
Hey guys! Let's dive into a matrix problem that often pops up in linear algebra. We're going to figure out when a product of two matrices doesn't have an inverse. It's like finding the sneaky values that make the matrix world a little unstable. So, let's get started!
Understanding the Problem
First, let's break down the question. We have two matrices, and we're multiplying them together. The big question is: when does this resulting matrix not have an inverse? Remember, a matrix has an inverse if and only if its determinant is non-zero. So, we're essentially looking for the value(s) of 'x' that make the determinant of the product matrix equal to zero. It's like finding the 'x' that causes a matrix meltdown!
Matrices and Invertibility: In the world of matrices, invertibility is super important. A matrix has an inverse if, and only if, its determinant isn't zero. Think of the determinant as a matrix's report card – if it scores zero, the matrix is a no-go for inversion. This concept is crucial for solving systems of equations, linear transformations, and a bunch of other stuff in linear algebra. So, when a matrix doesn't have an inverse, it means things can get a bit wonky in those applications. The question we're tackling asks us to find the exact value of 'x' that throws a wrench into the invertibility of the product matrix. We'll calculate the determinant of the resulting matrix after multiplying the two given matrices, set it equal to zero, and solve for 'x'. This value is the one that makes the matrix non-invertible, a key concept in linear algebra.
Matrix Multiplication
Let's start by multiplying the matrices. We have:
To multiply, we take the dot product of the rows of the first matrix with the columns of the second matrix. It's like a mathematical handshake between rows and columns!
- First Row, First Column: (3 * 2) + (2 * 3) = 6 + 6 = 12
- First Row, Second Column: (3 * -1) + (2 * 2) = -3 + 4 = 1
- Second Row, First Column: (x * 2) + (4 * 3) = 2x + 12
- Second Row, Second Column: (x * -1) + (4 * 2) = -x + 8
So, the product matrix is:
Step-by-Step Matrix Multiplication: To really nail matrix multiplication, think of it as a systematic process. You're essentially combining the rows of the first matrix with the columns of the second. The element in the first row and first column of the resulting matrix comes from multiplying the elements of the first row of the first matrix by the corresponding elements of the first column of the second matrix, and then adding those products together. You repeat this for each combination of rows and columns. So, for our matrices, we meticulously multiply and add: (3 * 2) + (2 * 3) for the top-left element, (3 * -1) + (2 * 2) for the top-right, (x * 2) + (4 * 3) for the bottom-left, and (x * -1) + (4 * 2) for the bottom-right. This careful process ensures we get the correct product matrix, which is crucial for the next step where we find the determinant. Trust me, getting this step right makes the rest of the problem way easier! We need this product matrix to figure out when it doesn't have an inverse.
Calculating the Determinant
Now, let's find the determinant of this product matrix. For a 2x2 matrix $egin{bmatrix} a & b \ c & d \end{bmatrix}$, the determinant is (a * d) - (b * c). It's like a simple formula that holds the key to invertibility!
So, for our matrix:
the determinant is:
(12 * (-x + 8)) - (1 * (2x + 12))
Let's simplify this:
-12x + 96 - 2x - 12
Combine like terms:
-14x + 84
The Magic of Determinants: The determinant is like a secret code that tells us a lot about a matrix. For a 2x2 matrix, it's calculated by subtracting the product of the off-diagonal elements from the product of the main diagonal elements. In our case, that's (12 * (-x + 8)) - (1 * (2x + 12)). This might look like just a bunch of numbers and variables, but it's a powerful expression. The determinant encapsulates crucial information about the matrix's properties, including whether it can be inverted. A non-zero determinant means the matrix has an inverse, which is super important for solving systems of linear equations and other applications. On the flip side, a determinant of zero means the matrix is singular and doesn't have an inverse, which can lead to some interesting and sometimes problematic situations. So, when we calculate -14x + 84, we're not just crunching numbers; we're unlocking a fundamental characteristic of the product matrix. This value will determine whether the matrix world stays stable or if it hits a snag because the inverse doesn't exist.
Setting the Determinant to Zero
To find when the matrix doesn't have an inverse, we set the determinant equal to zero:
-14x + 84 = 0
Now, let's solve for x:
-14x = -84
x = -84 / -14
x = 6
Finding the Critical Value: Setting the determinant to zero is like finding the exact tipping point where our matrix loses its invertibility powers. It's the value of 'x' that throws a wrench into the matrix's gears, making it singular and unable to be inverted. The equation -14x + 84 = 0 is where the magic happens. It represents the condition where the matrix's determinant vanishes, indicating a critical state. By solving for 'x', we're pinpointing the precise input that triggers this state. The steps are straightforward: isolate the term with 'x', then divide to find the value. This value, x = 6, is super important because it tells us exactly when our product matrix goes from being invertible to non-invertible. It's a key piece of information for understanding the behavior and properties of the matrix under different conditions.
The Answer
So, the product matrix does not have an inverse when x = 6. That's the value that makes the determinant zero, and thus, the matrix non-invertible. We nailed it!
Wrapping It Up: We've gone through the entire process step by step – multiplying matrices, calculating determinants, and finding the critical value of 'x'. We started with the fundamental idea that a matrix is invertible if its determinant isn't zero. Then, we multiplied the given matrices, carefully combining rows and columns to get the product matrix. Next, we calculated the determinant of this product, which gave us an expression involving 'x'. The real magic happened when we set this determinant equal to zero, because that's the condition for non-invertibility. Solving the resulting equation gave us x = 6, our final answer. This journey highlights the interconnectedness of these concepts in linear algebra. Each step builds on the previous one, leading us to a clear and concise solution. Understanding this process is key to tackling similar problems and grasping the broader concepts of matrix invertibility.
For what value of x will the product of the given matrices not have an inverse?
Matrix Invertibility Find the Value of x for No Inverse
