Determinant Of Matrix X Where AX = B + A^t

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Hey guys! Today, we're diving into a cool math problem involving matrices. We've got matrices A and B, and we need to figure out the determinant of matrix X, given the equation AX = B + A^t. Sounds like fun, right? Let's break it down step by step so we can all understand it clearly. This problem is a classic example of how matrix operations and determinants work together, and mastering it will definitely level up your linear algebra skills. So, grab your pencils, and let's get started!

1. Understanding the Problem

Okay, so first things first, let's make sure we all understand what's given. We have two matrices:

  • A = [[1, 2], [3, 5]]
  • B = [[3, -2], [1, 4]]

We also know that A^t is the transpose of matrix A. Remember, the transpose of a matrix is when you flip it over its main diagonal – basically, you swap the rows and columns. And the main equation we need to solve is AX = B + A^t. Our mission, should we choose to accept it (and we do!), is to find the determinant of matrix X. Determinants are super important in linear algebra because they tell us a lot about the matrix, like whether it's invertible or not. We need to understand each component of the problem statement. So, we need to find A transpose and understand the properties of matrix operations.

To really nail this, we need to be comfy with a few key concepts:

  • Matrix Transpose: Flipping rows and columns.
  • Matrix Addition: Adding matrices of the same dimensions.
  • Matrix Multiplication: Multiplying matrices (rows of the first by columns of the second).
  • Determinant of a Matrix: A special number calculated from a square matrix.

Let's look at each of these in a bit more detail:

  • Matrix Transpose: The transpose of a matrix A, denoted as A^t, is obtained by interchanging its rows into columns (or columns into rows). For example, if A = [[a, b], [c, d]], then A^t = [[a, c], [b, d]]. Transposing a matrix is a fundamental operation in linear algebra and is used in various applications, such as solving systems of linear equations and performing data analysis.

  • Matrix Addition: Matrix addition is performed by adding corresponding elements of two matrices of the same dimensions. If A = [[a, b], [c, d]] and B = [[e, f], [g, h]], then A + B = [[a+e, b+f], [c+g, d+h]]. Matrix addition follows the commutative and associative properties, making it a straightforward operation for combining matrices.

  • Matrix Multiplication: Matrix multiplication is a more complex operation compared to addition. To multiply two matrices A and B, the number of columns in A must be equal to the number of rows in B. If A is an m x n matrix and B is an n x p matrix, then the resulting matrix AB will be an m x p matrix. The elements of the product matrix are calculated by taking the dot product of the rows of A and the columns of B. For example, if A = [[a, b], [c, d]] and B = [[e, f], [g, h]], then AB = [[ae+bg, af+bh], [ce+dg, cf+dh]]. Matrix multiplication is not commutative in general, meaning AB ≠ BA.

  • Determinant of a Matrix: The determinant of a matrix is a scalar value that can be computed from the elements of a square matrix. For a 2x2 matrix A = [[a, b], [c, d]], the determinant, denoted as det(A) or |A|, is calculated as ad - bc. The determinant provides important information about the matrix, such as whether the matrix is invertible (if the determinant is non-zero) and the volume scaling factor of the linear transformation represented by the matrix. Determinants play a crucial role in solving linear equations, finding eigenvalues, and performing various matrix operations.

2. Finding A Transpose (A^t)

Alright, let's get practical. First up, we need to find A^t. Remember, this means we swap the rows and columns of A. So, if A = [[1, 2], [3, 5]], then A^t becomes:

A^t = [[1, 3], [2, 5]]

See? The first row of A (1, 2) becomes the first column of A^t, and the second row of A (3, 5) becomes the second column of A^t. Easy peasy!

Calculating the transpose is a fundamental step in many matrix operations. It's like a mirror reflection across the main diagonal. This might seem simple, but it's super important. The transpose helps us in several applications, such as solving systems of linear equations, finding eigenvalues, and even in computer graphics for transformations. So, mastering this little trick is a big win for our math toolkit.

3. Calculating B + A^t

Next up, we need to figure out B + A^t. This means we're adding two matrices together. Remember, to add matrices, they need to have the same dimensions (which A^t and B do), and we just add the corresponding elements. So:

B + A^t = [[3, -2], [1, 4]] + [[1, 3], [2, 5]] = [[3+1, -2+3], [1+2, 4+5]] = [[4, 1], [3, 9]]

We just added each element in B to its corresponding element in A^t. See how the top-left element of B (3) is added to the top-left element of A^t (1) to get 4, and so on? This is matrix addition in a nutshell. Matrix addition is a foundational operation, similar to adding numbers, but with a grid structure. It's commutative and associative, which means the order in which we add matrices doesn't change the result (A + B = B + A) and (A + B) + C = A + (B + C). Understanding these properties can make complex calculations much simpler. Matrix addition is essential in various fields like computer graphics, physics, and engineering, where it's used for combining transformations and forces.

4. Solving for X: AX = B + A^t

Now comes the slightly trickier part. We have the equation AX = B + A^t, and we want to find X. We already know B + A^t, so we can rewrite the equation as AX = [[4, 1], [3, 9]]. To isolate X, we need to get rid of A on the left side. The way we do this in matrix algebra is by multiplying both sides by the inverse of A (if it exists). Let's call the inverse of A as A^(-1).

So, we get:

A^(-1)AX = A^(-1)([[4, 1], [3, 9]])

Since A^(-1)A is the identity matrix (a matrix with 1s on the diagonal and 0s everywhere else), it's like multiplying by 1, so we have:

X = A^(-1)([[4, 1], [3, 9]])

This means we need to find the inverse of A and then multiply it by the matrix [[4, 1], [3, 9]].

4.1 Finding the Inverse of A (A^(-1))

For a 2x2 matrix like A = [[1, 2], [3, 5]], the inverse is calculated as follows:

  1. Find the determinant of A (det(A)).
  2. Swap the elements on the main diagonal (top-left and bottom-right).
  3. Change the signs of the off-diagonal elements (top-right and bottom-left).
  4. Divide the resulting matrix by the determinant.

So, let's do it:

  1. det(A) = (1 * 5) - (2 * 3) = 5 - 6 = -1
  2. Swap the diagonal elements: [[5, 2], [3, 1]]
  3. Change the signs of the off-diagonal elements: [[5, -2], [-3, 1]]
  4. Divide by the determinant (-1): A^(-1) = [[-5, 2], [3, -1]]

Now we have the inverse of A! Finding the inverse of a matrix is a crucial step in solving many matrix equations. It's like division in regular algebra, but with matrices. However, not all matrices have inverses. A matrix must be square (same number of rows and columns) and have a non-zero determinant to be invertible. The inverse of a matrix, when multiplied by the original matrix, gives the identity matrix, which acts as the '1' in matrix multiplication. This property makes the inverse incredibly useful for solving systems of linear equations and performing various transformations in fields like computer graphics and engineering.

4.2 Calculating X

Now we can find X by multiplying A^(-1) by [[4, 1], [3, 9]]:

X = [[-5, 2], [3, -1]] * [[4, 1], [3, 9]]

To multiply these matrices, we take the dot product of the rows of the first matrix and the columns of the second matrix:

X = [[(-54) + (23), (-51) + (29)], [(34) + (-13), (31) + (-19)]] X = [[-20 + 6, -5 + 18], [12 - 3, 3 - 9]] X = [[-14, 13], [9, -6]]

So, matrix X is [[-14, 13], [9, -6]].

5. Finding the Determinant of X

Finally, we're at the last step! We need to find the determinant of X. For a 2x2 matrix, it's (top-left * bottom-right) - (top-right * bottom-left):

det(X) = (-14 * -6) - (13 * 9) det(X) = 84 - 117 det(X) = -33

And there we have it! The determinant of matrix X is -33.

Conclusion

So, we walked through the whole problem step by step. We started by understanding the problem and the key concepts, then we found A^t, calculated B + A^t, solved for X by finding A^(-1) and multiplying, and finally, we calculated the determinant of X. Phew! That was a journey, but we made it. Remember, the key to mastering these problems is to break them down into smaller, manageable steps. Keep practicing, and you'll become a matrix whiz in no time! And, of course, if you have any questions or want to try another problem, just let me know. Keep up the awesome work, guys! Determinants are a cornerstone of linear algebra. They tell us crucial information about matrices, like whether a matrix is invertible and the volume scaling factor of a linear transformation. A zero determinant indicates a singular matrix (non-invertible), while a non-zero determinant means the matrix has an inverse. This property is vital in solving systems of linear equations. The sign of the determinant can also tell us about the orientation of the transformation, such as whether it involves a reflection. In higher dimensions, the determinant gives the volume scaling factor of the transformation, making it a key concept in multivariable calculus and physics.