Solve Matrix Equations: Find B In AB = C
Hey guys! Today, we're diving deep into the fascinating world of matrix operations, specifically focusing on solving matrix equations. It might sound intimidating, but trust me, with a step-by-step approach and a sprinkle of practice, you'll be a pro in no time! We'll be dissecting a problem where we're given a matrix A and the product of matrices A and B (AB), and our mission, should we choose to accept it, is to find the elusive matrix B. So, buckle up, grab your thinking caps, and let's get started!
Unveiling the Matrix Equation: A Step-by-Step Approach
The core of our journey lies in understanding the fundamental principles of matrix algebra. Before we jump into the solution, let's break down the key concepts and techniques we'll be using. Understanding these concepts is crucial, guys, because they form the bedrock of all matrix operations. We'll cover matrix multiplication, the concept of an inverse matrix, and how these tools can be used to solve for unknown matrices in equations. Think of it like learning the grammar of a new language – once you've got the basics down, you can start forming complex sentences (or in our case, solving complex matrix equations!).
Understanding Matrix Multiplication
Matrix multiplication isn't just your regular number multiplication; it's a whole different ball game. To multiply two matrices, the number of columns in the first matrix must equal the number of rows in the second matrix. The resulting matrix will have the same number of rows as the first matrix and the same number of columns as the second matrix. The elements of the resulting matrix are calculated by taking the dot product of the rows of the first matrix and the columns of the second matrix. This might sound like a mouthful, but it's actually quite straightforward once you get the hang of it. Imagine each element in the resulting matrix as a meeting point between a row from the first matrix and a column from the second matrix. The dot product is simply a way of combining the information from that row and column to determine the value of that element. For example, if we have matrix A with dimensions m x n and matrix B with dimensions n x p, the resulting matrix AB will have dimensions m x p. This is a crucial concept, so make sure you've got it down pat before moving on!
The Power of the Inverse Matrix
Now, let's talk about the inverse matrix. Not every matrix has an inverse, but when it does, it's like the magic key to solving matrix equations. The inverse of a matrix, denoted as A⁻¹, is a matrix that, when multiplied by the original matrix A, results in the identity matrix (I). The identity matrix is like the number 1 in regular multiplication; it doesn't change the matrix when multiplied. Finding the inverse of a matrix involves several steps, including calculating the determinant and the adjugate matrix. The determinant is a scalar value that can be calculated from the elements of the matrix, and the adjugate matrix is the transpose of the matrix of cofactors. The inverse is calculated by dividing the adjugate matrix by the determinant. This process is a bit involved, but there are plenty of online calculators and resources that can help you out. The key takeaway here is that the inverse matrix allows us to "undo" the multiplication by the original matrix, which is exactly what we need to solve our equation.
Solving for the Unknown Matrix
With these tools in our arsenal, we can now tackle the challenge of solving for the unknown matrix. In our case, we have the equation AB = C, where A and C are known matrices, and we want to find B. To isolate B, we can multiply both sides of the equation by the inverse of A (A⁻¹), but here's the catch: matrix multiplication is not commutative, meaning the order matters. We need to multiply A⁻¹ on the left side of both AB and C. This gives us A⁻¹AB = A⁻¹C. Since A⁻¹A equals the identity matrix I, we have IB = A⁻¹C, and since multiplying by the identity matrix doesn't change the matrix, we finally get B = A⁻¹C. This is the magic formula! We've successfully isolated B, and all that's left is to calculate the inverse of A and perform the matrix multiplication.
Cracking the Code: Solving the Specific Problem
Alright, guys, let's get down to business and apply these concepts to our specific problem. We are given:
A =
egin{bmatrix}
1 & 0 & 0 \
0 & -1 & 0 \
0 & 0 & 1
and
AB =
egin{bmatrix}
-1 & 0 & 0 \
0 & -1 & 0 \
0 & 0 & -1
## Step 1: Finding the Inverse of Matrix A
The first crucial step in solving for matrix B is to **find the inverse of matrix A**. But before we dive into the calculation, let's recap why this is so important. As we discussed earlier, the inverse of a matrix acts like the "undo" button for matrix multiplication. By multiplying both sides of the equation AB = C by A⁻¹, we can isolate B and get closer to our solution. Now, let's get our hands dirty and calculate A⁻¹!
### Calculating the Determinant
To find the inverse of a matrix, we first need to calculate its determinant. The determinant is a special scalar value that provides crucial information about the matrix, including whether it has an inverse. For a 3x3 matrix like our A, the determinant can be calculated using a specific formula. There are a couple of ways to do this, but the most common method involves expanding along a row or column. For our matrix A, we can use the first row, since it has two zeros, making the calculation easier. The formula involves multiplying each element in the first row by the determinant of its corresponding 2x2 minor matrix (the matrix formed by deleting the row and column of that element), and then alternating signs. Don't worry if this sounds complicated; it's much easier in practice!
Let's break it down: for our matrix A, the determinant is calculated as (1 * determinant of the 2x2 matrix formed by deleting the first row and first column) - (0 * determinant of its corresponding minor) + (0 * determinant of its corresponding minor). Since the last two terms are multiplied by zero, we only need to calculate the determinant of the 2x2 matrix. This 2x2 matrix is:
[[-1, 0], [0, 1]]
The determinant of a 2x2 matrix is calculated by multiplying the top-left and bottom-right elements and subtracting the product of the top-right and bottom-left elements. In this case, it's (-1 * 1) - (0 * 0) = -1. Therefore, the determinant of matrix A is (1 * -1) = -1. **This is a key value**, as it will be used in the next step to calculate the adjugate matrix.
### Finding the Adjugate Matrix
The next step in finding the inverse is to calculate the adjugate matrix. The adjugate matrix, also known as the adjoint matrix, is the transpose of the matrix of cofactors. Let's break this down step by step. First, we need to find the matrix of cofactors. A cofactor is a signed minor. The minor of an element is the determinant of the submatrix formed by deleting the row and column containing that element. The sign of the cofactor is determined by a checkerboard pattern of + and - signs. For a 3x3 matrix, the pattern is:
[+, -, +], [-, +, -], [+, -, +]
So, to find the cofactor of an element, we calculate its minor, multiply it by the appropriate sign from the checkerboard pattern, and that's our cofactor. Once we've calculated all the cofactors, we arrange them in a matrix. This is our matrix of cofactors. Finally, to get the adjugate matrix, we simply take the transpose of the matrix of cofactors, which means swapping its rows and columns. **It's a bit of a process**, but with practice, it becomes second nature.
For matrix A, let's walk through the cofactor calculation. The cofactor of the element in the first row and first column (which is 1) is the determinant of the 2x2 matrix we calculated earlier, multiplied by +1 (from the checkerboard pattern), which is -1. The cofactor of the element in the first row and second column (which is 0) is the negative of the determinant of the 2x2 matrix formed by deleting the first row and second column, which is -(0 * 1 - 0 * 0) = 0. Similarly, the cofactor of the element in the first row and third column (which is 0) is 0. Continuing this process for all elements, we get the matrix of cofactors:
[[-1, 0, 0], [0, 1, 0], [0, 0, -1]]
Taking the transpose of this matrix, we get the adjugate matrix:
[[-1, 0, 0], [0, 1, 0], [0, 0, -1]]
In this case, the matrix of cofactors and the adjugate matrix are the same, but this isn't always the case!
### Calculating the Inverse Matrix
Now that we have the determinant and the adjugate matrix, we can finally calculate the inverse matrix. The formula for the inverse is A⁻¹ = (1 / determinant of A) * adjugate of A. In our case, the determinant of A is -1, and the adjugate of A is the matrix we just calculated. So, we simply multiply each element of the adjugate matrix by (1 / -1) = -1. This gives us the inverse matrix:
A⁻¹ = [1 & 0 & 0 \ 0 & -1 & 0 \ 0 & 0 & 1]
**Interestingly, in this case, the inverse of A is the same as A itself!** This is a special property of certain matrices, known as involutory matrices.
## Step 2: Multiplying by the Inverse
Now that we've successfully navigated the tricky terrain of finding the inverse, it's time to put it to good use. Remember our goal: to isolate matrix B in the equation AB = C. We've already established that multiplying both sides of the equation by A⁻¹ on the left will do the trick. So, let's do it!
We have the equation AB = C, where:
A = [1 & 0 & 0 \ 0 & -1 & 0 \ 0 & 0 & 1]
AB = C = [-1 & 0 & 0 \ 0 & -1 & 0 \ 0 & 0 & -1]
And we've calculated A⁻¹ to be:
A⁻¹ = [1 & 0 & 0 \ 0 & -1 & 0 \ 0 & 0 & 1]
Multiplying both sides of AB = C by A⁻¹ on the left, we get A⁻¹AB = A⁻¹C. As we discussed earlier, A⁻¹A equals the identity matrix I, so we have IB = A⁻¹C, which simplifies to B = A⁻¹C. Now, all that's left is to perform the matrix multiplication A⁻¹C. **This is the final hurdle**, guys, so let's power through!
### Performing the Matrix Multiplication
To multiply two matrices, we take the dot product of the rows of the first matrix and the columns of the second matrix. Let's break it down element by element. The element in the first row and first column of the resulting matrix B is the dot product of the first row of A⁻¹ and the first column of C. This is (1 * -1) + (0 * 0) + (0 * 0) = -1. Similarly, the element in the first row and second column of B is the dot product of the first row of A⁻¹ and the second column of C, which is (1 * 0) + (0 * -1) + (0 * 0) = 0. We continue this process for all elements of the resulting matrix.
After performing the matrix multiplication, we get:
B = A⁻¹C = [-1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & -1]
## The Grand Finale: Identifying the Matrix Category
Woohoo! We've successfully solved for matrix B! But our journey doesn't end here. The final part of the problem asks us to identify the category to which matrix B belongs. This requires us to take a closer look at the properties of matrix B and see if it fits any specific classifications. **This is where our understanding of different matrix types comes into play.**
### Analyzing Matrix B
Let's examine matrix B:
B = [-1 & 0 & 0 \ 0 & 1 & 0 \ 0 & 0 & -1]
Notice that B is a diagonal matrix, meaning all the elements outside the main diagonal (the diagonal from the top-left to the bottom-right) are zero. This is a good starting point, but it's not the whole story. We need to delve deeper and see if it fits any more specific categories.
### Recognizing Reflection Matrices
Looking closely at the diagonal elements, we see that two of them are -1 and one is 1. This pattern is a telltale sign of a **reflection matrix**. A reflection matrix is a linear transformation that reflects vectors across a plane or a line. In 3D space, a reflection matrix typically has two -1s and one 1 on the diagonal (or vice versa), and it represents a reflection across a plane. In our case, matrix B represents a reflection across the yz-plane (the plane defined by the y and z axes), since the x-coordinate changes sign while the y and z coordinates remain unchanged. **This is the key insight!**
## Conclusion: Mastering Matrix Operations
And there you have it, guys! We've successfully navigated the world of matrix equations, solved for an unknown matrix, and even identified its category. We've covered a lot of ground, from the fundamentals of matrix multiplication and inverse matrices to the intricacies of reflection matrices. The key to mastering matrix operations is practice, practice, practice! The more you work with matrices, the more comfortable you'll become with the concepts and techniques. So, keep practicing, keep exploring, and keep pushing your mathematical boundaries. You've got this!
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