Matrix Subtraction: Finding C - D And Simplifying

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Hey guys! Let's dive into a matrix subtraction problem. We're given two matrices, C and D, and our mission is to find C - D and simplify the result if we can. Matrix operations can be super useful in various fields like computer graphics, data analysis, and engineering. So, let's break it down step by step.

Understanding the Matrices

First, let's clearly define our matrices. We have matrix C:

C=[1121−1417]C=\left[\begin{array}{cc}11 & 2 \\ 1 & -\frac{1}{4} \\ 1 & 7\end{array}\right]

And matrix D:

D=[422−2213]D=\left[\begin{array}{ccc}4 & 2 & 2 \\ -2 & 2 & \frac{1}{3}\end{array}\right]

To subtract matrices, they must have the same dimensions. That's a crucial rule! Matrix C is a 3x2 matrix (3 rows and 2 columns), while matrix D is a 2x3 matrix (2 rows and 3 columns). Because their dimensions don't match, we cannot perform the subtraction C - D directly. Therefore, C-D is undefined.

Dimensions and Compatibility

The dimensions of a matrix are given by the number of rows and columns it has. For instance, a matrix with m rows and n columns is an m x n matrix. In order to add or subtract two matrices, they need to have the exact same number of rows and columns. This requirement ensures that each element in the first matrix has a corresponding element in the second matrix to perform the operation. If the dimensions don't match, the operation is undefined.

For example, consider two matrices A and B:

A=[a11a12a21a22]A = \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix}

B=[b11b12b21b22]B = \begin{bmatrix} b_{11} & b_{12} \\ b_{21} & b_{22} \end{bmatrix}

Both matrices A and B are 2x2 matrices. Therefore, we can perform addition or subtraction element-wise. The resulting matrix C would be:

C=A+B=[a11+b11a12+b12a21+b21a22+b22]C = A + B = \begin{bmatrix} a_{11} + b_{11} & a_{12} + b_{12} \\ a_{21} + b_{21} & a_{22} + b_{22} \end{bmatrix}

However, if we had a matrix D that was a 2x3 matrix:

D=[d11d12d13d21d22d23]D = \begin{bmatrix} d_{11} & d_{12} & d_{13} \\ d_{21} & d_{22} & d_{23} \end{bmatrix}

Then, we could not add or subtract matrix A from matrix D because they have different dimensions. Matrix A is 2x2 and Matrix D is 2x3.

Why Dimensions Matter

The reason dimensions matter is that matrix addition and subtraction are defined element-wise. You add or subtract corresponding elements in the matrices. If the matrices have different dimensions, there won't be a corresponding element for every element in the matrix, making the operation impossible. This requirement ensures that the resulting matrix has a well-defined structure and maintains the mathematical properties of matrix operations.

In fields like computer graphics, matrices are often used to represent transformations of geometric objects. Performing operations on matrices with mismatched dimensions would lead to incorrect transformations and visual artifacts. Similarly, in data analysis, matrices are used to store and manipulate data sets. Consistent dimensions are crucial for performing meaningful statistical operations and ensuring data integrity.

Checking Dimensions

The key to determining whether we can subtract these matrices lies in their dimensions. Remember, matrix C is a 3x2 matrix, and matrix D is a 2x3 matrix. To perform any addition or subtraction, the matrices must have identical dimensions.

What are Matrix Dimensions?

When we talk about matrix dimensions, we are referring to the number of rows and columns a matrix has. The dimension is always written as rows x columns. So, a 3x2 matrix has 3 rows and 2 columns, while a 2x3 matrix has 2 rows and 3 columns.

Understanding matrix dimensions is fundamental to performing matrix operations. It's like knowing the units of measurement when you're doing physics problems; you can't add meters and seconds! Similarly, you can't add or subtract matrices if their dimensions don't match.

Why Dimensions Matter for Subtraction

Matrix subtraction is performed element-wise. This means that you subtract the element in the i-th row and j-th column of the second matrix from the corresponding element in the i-th row and j-th column of the first matrix. For example, if you have two 2x2 matrices A and B:

A=[a11a12a21a22]A = \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{bmatrix} and B=[b11b12b21b22]B = \begin{bmatrix} b_{11} & b_{12} \\ b_{21} & b_{22} \end{bmatrix}

Then, A - B would be:

A−B=[a11−b11a12−b12a21−b21a22−b22]A - B = \begin{bmatrix} a_{11} - b_{11} & a_{12} - b_{12} \\ a_{21} - b_{21} & a_{22} - b_{22} \end{bmatrix}

Notice that each element in A is being subtracted by the corresponding element in B. If the matrices have different dimensions, this element-wise subtraction becomes impossible because not every element in the first matrix will have a corresponding element in the second matrix.

Implications of Mismatched Dimensions

When you attempt to subtract matrices with mismatched dimensions, you will quickly find that you run out of corresponding elements. For instance, in our case, matrix C is 3x2 and matrix D is 2x3. If we tried to subtract D from C, we would have:

C−D=[1121−1417]−[422−2213]C - D = \begin{bmatrix} 11 & 2 \\ 1 & -\frac{1}{4} \\ 1 & 7 \end{bmatrix} - \begin{bmatrix} 4 & 2 & 2 \\ -2 & 2 & \frac{1}{3} \end{bmatrix}

We can see that the first row of C (11, 2) can be subtracted by the first row of D (4, 2, 2), but then we run into a problem. C only has two columns, while D has three in its first row. This mismatch continues for the other rows, making the subtraction undefined.

Real-World Analogy

Think of it like trying to subtract two images of different sizes pixel by pixel. If one image is 100x100 pixels and the other is 50x50 pixels, you can't directly subtract them because they don't align. You would need to resize or crop one of the images first to make their dimensions match.

Similarly, in matrix operations, the dimensions must match to ensure that the operation is well-defined and mathematically sound. This requirement is crucial in various applications, including computer graphics, data analysis, and machine learning, where matrices are used extensively.

Conclusion

In summary, because the dimensions of matrix C (3x2) and matrix D (2x3) are different, the operation C - D is not defined. Always remember to check the dimensions of matrices before attempting to add or subtract them! Matrix operations rely heavily on these rules, so understanding them is super important for your mathematical journey!