Problem 106 - 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 \(\quad B=\left[\begin{array}{rr}-1 & 1 \\ 0 & -2\end{array}\right].\) Show that \((A+B)^{2}=A^{2}+A B+B A+B^{2}\)

Step-by-Step Solution

Answer

By calculating both sides of the equation \((A+B)²=A²+AB+BA+B²\), it is shown that they are equal, verifying the given equation.

Step 1: Prepare Matrices

First, write down the Matrices A and B as given in the exercise: \(A=\left[\begin{array}{rr}2 & -1 \ 1 & 3\end{array}\right]\) and \(B=\left[\begin{array}{rr}-1 & 1 \ 0 & -2\end{array}\right]\)

Step 2: Perform (A+B)²

Calculate \(A+B\) by adding corresponding elements in the matrices A and B. The result will be a new matrix. Multiply the resulting matrix by itself to get \((A+B)²\)

Step 3: Perform A²+AB+BA+B²

Calculate the square of matrix A by multiplying matrix A with itself, calculate the product of matrix A and matrix B, the product of matrix B and matrix A, and the square of matrix B by multiplying matrix B with itself. Then add up all the resulting matrices. This will give you \(A²+AB+BA+B²\)

Step 4: Compare results

Lastly, compare the results from step 2 and step 3. If they are the same, then the equation \((A+B)²=A²+AB+BA+B²\) is verified

Key Concepts

Matrix Addition

Matrix addition is a simple operation where corresponding elements from two matrices are summed to produce a new matrix. To perform this operation, the matrices must be of the same dimensions, meaning they have the same number of rows and columns. Consider matrices A and B, if both are 2x2 matrices, you can add them by adding each element in A with the corresponding element in B.
For example, given matrices:

• Matrix A: \[\begin{bmatrix}2 & -1 \1 & 3\end{bmatrix}\]
• Matrix B: \[\begin{bmatrix}-1 & 1 \0 & -2\end{bmatrix}\]

The sum A+B is: \[\begin{bmatrix}2+(-1) & -1+1 \1+0 & 3+(-2)\end{bmatrix} = \begin{bmatrix}1 & 0 \1 & 1\end{bmatrix}\]This new matrix is the result of adding A and B.

Matrix Multiplication

Matrix multiplication is slightly more complex than addition. It involves the product of two matrices, which is achieved by taking the dot product of rows and columns. For multiplication to be possible, the number of columns in the first matrix must match the number of rows in the second matrix.
For example, if you have matrices A and B both of size 2x2, you can multiply them as follows:

• Take the first row of A and multiply it by the first column of B, then sum the products to get the element in the first row, first column of the result matrix.
• Continue this process using different combinations of rows from the first matrix and columns from the second matrix.

Here's a representation, when A = \[\begin{bmatrix}2 & -1 \1 & 3\end{bmatrix}\], and B = \[\begin{bmatrix}-1 & 1 \0 & -2\end{bmatrix}\], A \( \cdot \) B is calculated as follows: \[\begin{bmatrix}2*(-1) + (-1)*0 & 2*1 + (-1)*(-2) \1*(-1) + 3*0 & 1*1 + 3*(-2)\end{bmatrix} = \begin{bmatrix}-2 & 4 \-1 & -5\end{bmatrix}\]This describes the resulting matrix from the multiplication of A and B.

Properties of Matrices

Understanding the properties of matrices is crucial for operations like addition and multiplication. One important property is the commutative property, which applies to addition but not always to multiplication.
For matrix addition, the sum of A and B is the same as the sum of B and A, i.e., \( A + B = B + A \). However, for multiplication, \( A \cdot B \) is not necessarily equal to \( B \cdot A \).

• Another key property to consider is the distributive property, which is applicable in equations like \((A+B)^2\).
• It states that \( (A+B)^2 = A^2 + AB + BA + B^2 \), illustrating how terms are distributed over the matrices.

These properties help simplify complex matrix expressions and allow for verification of equations such as \( (A+B)^2 = A^2 + AB + BA + B^2 \). Understanding these properties can aid in ensuring calculations are performed correctly.

Matrix Equality

Matrix equality is a concept that describes when two matrices are considered equal. For two matrices to be equal, they must have the same dimensions, and every corresponding element within the matrices must be identical.
This means for matrices A and B to be equal, every element \( a_{ij} \) in matrix A must be equal to the corresponding element \( b_{ij} \) in matrix B. In mathematical terms, matrix A of dimension m×n is equal to matrix B of the same dimension if \( a_{ij} = b_{ij} \) for all i, j.

• If two matrices are equal, then their addition or multiplication with another matrix will yield the same result for both.
• In context of proving an equation with matrices, it is necessary to verify equality through comparison of the resultant matrices obtained from different operations.

In exercises like \((A+B)^2 = A^2 + AB + BA + B^2\), comparing the results side by side ensures you are working with equivalent matrices, verifying the equation's validity.