Problem 104 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use the matrices \(A=\left[\begin{array}{rr}2 & -1 \\ 1 & 3\end{array}\right] \quad\) and $\quad B=\left[\begin{array}{rr}-1 & 1 \\ 0 & -2\end{array}\right].$$$\text { Show that }(A-B)^{2} \neq A^{2}-2 A B+B^{2}$$

Step-by-Step Solution

Answer

\((A-B)^2\) is not equal to \(A^2 - 2AB + B^2\) for the given matrices. This confirms that square of difference is not equal to difference of squares in the case of matrices.

Step 1: Subtraction of the Matrices

Subtract matrix B from A to get the matrix (A-B). That is,\n \(A-B = \left[\begin{array}{rr}2 & -1 \ 1 & 3\end{array}\right] - \left[\begin{array}{rr}-1 & 1 \ 0 & -2\end{array}\right] = \left[\begin{array}{rr}3 & -2 \ 1 & 5\end{array}\right]\)

Step 2: Squaring the Resulted Matrix

Calculate the square of the matrix (A-B) by multiplying it with itself. That is, \((A-B)^2 = \left[\begin{array}{rr}3 & -2 \ 1 & 5\end{array}\right] * \left[\begin{array}{rr}3 & -2 \ 1 & 5\end{array}\right] = \left[\begin{array}{rr}5 & -16 \ 8 & 21\end{array}\right]\)

Step 3: Calculating Individual Square of Matrices

Calculate the individual square of the matrices A and B, i.e., calculate \(A^2\) and \(B^2\). That is,\n \(A^2 = \left[\begin{array}{rr}2 & -1 \ 1 & 3\end{array}\right] * \left[\begin{array}{rr}2 & -1 \ 1 & 3\end{array}\right] = \left[\begin{array}{rr}3 & -7 \ 5 & 8\end{array}\right]\) and \n \(B^2 = \left[\begin{array}{rr}-1 & 1 \ 0 & -2\end{array}\right] * \left[\begin{array}{rr}-1 & 1 \ 0 & -2\end{array}\right] = \left[\begin{array}{rr}-1 & -2 \ 0 & 4\end{array}\right]\)

Step 4: Calculating Product of A and B

Calculate the product of A and B, i.e., calculate \(AB\). That is,\n \(AB = \left[\begin{array}{rr}2 & -1 \ 1 & 3\end{array}\right] * \left[\begin{array}{rr}-1 & 1 \ 0 & -2\end{array}\right] = \left[\begin{array}{rr}-2 & 0 \ 1 & -5\end{array}\right]\)

Step 5: Check the Equality

Finally, for checking the condition given in the original problem statement, specifically if \(A^2 - 2AB + B^2\) equals \((A-B)^2\), calculate \(A^2 - 2AB + B^2\). That is,\n \(A^2 - 2AB + B^2 = \left[\begin{array}{rr}3 & -7 \ 5 & 8\end{array}\right] - 2*\left[\begin{array}{rr}-2 & 0 \ 1 & -5\end{array}\right] + \left[\begin{array}{rr}-1 & -2 \ 0 & 4\end{array}\right]\) = \left[\begin{array}{rr}5 & -7 \ 6 & 22\end{array}\right]\). Clearly, \( (A-B)^2 \neq A^2 - 2AB + B^2\), thereby confirming that the object equality doesn't hold true for matrices.

Key Concepts

Matrix Subtraction

Matrix subtraction is an operation where each element of one matrix is subtracted from the corresponding element of another matrix. To subtract one matrix from another, both matrices must have the same dimensions. Imagine you have two matrices, Matrix A and Matrix B, the subtraction A - B results in a new matrix where each entry c_{ij} is the difference of entries a_{ij} and b_{ij} from matrices A and B respectively.

For example, if we subtract matrix B from matrix A (A - B), we perform the subtraction element-wise:
\[ A - B = \left[\begin{array}{cc} a_{11}-b_{11} & a_{12}-b_{12} \ a_{21}-b_{21} & a_{22}-b_{22} \end{array}\right] \].
This operation doesn't simply negate the properties of matrix addition; as we'll see with matrix multiplication, subtraction can lead to results that are non-intuitive, especially when combined with other matrix operations.

Square of a Matrix

The square of a matrix, not to be confused with element-wise squaring, involves multiplying a matrix by itself. It's denoted as A² for a matrix A. For a matrix to be squared, it must be a square matrix itself, meaning it has the same number of rows and columns. This process is done through the standard matrix multiplication rules.

Standard Matrix Multiplication

To calculate A², each entry (i, j) of the resulting matrix is computed as the sum of the products of the corresponding elements from the ith row of the first matrix and the jth column of the second matrix. This can be visually understood through the dot product of rows and columns in the multiplication process.

It's crucial to comprehend that the square of a matrix is not simply taking each entry and squaring it; it is a more involved process that includes combining elements from both matrices in a specific way.

Matrix Algebra

Matrix algebra encompasses the rules and operations that can be performed on matrices, including addition, subtraction, multiplication, and scalar multiplication. Unlike regular algebra, matrix algebra does not always behave in intuitive ways, especially when it concerns multiplication and subtraction.

Non-Commutativity

One unique property of matrix algebra is the non-commutative nature of matrix multiplication; meaning, in general, AB ≠ BA. Each operation in matrix algebra follows specific rules, for example, there are conditions for when certain operations can be performed—such as the requirement for matrices to be of compatible dimensions for addition and multiplication.

Understanding the nuances of these operations is essential for solving matrix equations and performing complex calculations found in linear algebra and related fields.

Properties of Matrix Operations

Matrix operations adhere to specific properties that dictate how they can be manipulated within equations. These properties include:

• Associativity of addition and multiplication ((A + B) + C = A + (B + C), (AB)C = A(BC)).
• Commutativity of addition, but not multiplication (A + B = B + A, AB ≠ BA typically).
• Distributive property (A(B + C) = AB + AC, likewise with (B + C)A = BA + CA).

The exercise depicted in the problem shows a case where these properties are applied to demonstrate that equations of matrices do not necessarily follow the same rules as equations with real numbers. In particular, when squaring a matrix after subtraction, the result is not equivalent to the square of the individual matrices minus twice the product of the matrices. This discrepancy underscores the importance of understanding the distinct properties of matrix operations, something essential for advancing in studies involving linear algebra and its applications.