Problem 2 - Algebra for College Students

Question

For Problems \(1-8\), find \(A+B, A-B, 2 A+3 B\), and \(4 A-2 B\). $$ A=\left[\begin{array}{rr} 3 & -6 \\ 2 & -1 \\ -4 & 5 \end{array}\right], \quad B=\left[\begin{array}{rr} 1 & 0 \\ 5 & -7 \\ -6 & 9 \end{array}\right] $$

Step-by-Step Solution

Answer

\(A+B = \begin{bmatrix} 4 & -6 \\ 7 & -8 \\ -10 & 14 \end{bmatrix}, A-B = \begin{bmatrix} 2 & -6 \\ -3 & 6 \\ 2 & -4 \end{bmatrix}, 2A+3B = \begin{bmatrix} 9 & -12 \\ 19 & -23 \\ -26 & 37 \end{bmatrix}, 4A-2B = \begin{bmatrix} 10 & -24 \\ -2 & 10 \\ -4 & 2 \end{bmatrix}."

Step 1: Matrix Addition A+B

To find \(A + B\), add corresponding elements in matrices \(A\) and \(B\).\[3+1 = 4,\ -6+0 = -6\2+5 = 7,\ -1 -7 = -8\-4-6 = -10,\ 5+9 = 14\]Thus, \(A+B = \left[\begin{array}{rr}4 & -6 \7 & -8 \-10 & 14\end{array}\right]\).

Step 2: Matrix Subtraction A-B

To find \(A - B\), subtract corresponding elements in matrix \(B\) from matrix \(A\).\[3-1 = 2,\ -6-0 = -6\2-5 = -3,\ -1+7 = 6\-4+6 = 2,\ 5-9 = -4\]Thus, \(A-B = \left[\begin{array}{rr}2 & -6 \-3 & 6 \2 & -4\end{array}\right]\).

Step 3: Scaling and Adding Matrices 2A+3B

To find \(2A + 3B\), first multiply matrix \(A\) by 2 and matrix \(B\) by 3, then add the results.\[2A = \left[ \begin{array}{rr}6 & -12 \4 & -2 \-8 & 10\end{array} \right], \ 3B = \left[ \begin{array}{rr}3 & 0 \15 & -21 \-18 & 27\end{array} \right]\]Now add the resulting matrices:\[6+3 = 9,\ -12+0 = -12\4+15 = 19,\ -2-21 = -23\-8-18 = -26,\ 10+27 = 37\]Thus, \(2A + 3B = \left[ \begin{array}{rr}9 & -12 \19 & -23 \-26 & 37\end{array} \right]\).

Step 4: Scaling and Subtracting Matrices 4A-2B

To find \(4A - 2B\), first multiply matrix \(A\) by 4 and matrix \(B\) by 2, then subtract the scaled \(B\) from the scaled \(A\).\[4A = \left[ \begin{array}{rr}12 & -24 \8 & -4 \-16 & 20\end{array} \right], \ 2B = \left[ \begin{array}{rr}2 & 0 \10 & -14 \-12 & 18\end{array} \right]\]Subtract the resulting matrices:\[12-2 = 10,\ -24-0 = -24\8-10 = -2,\ -4+14 = 10\-16+12 = -4,\ 20-18 = 2\]Thus, \(4A - 2B = \left[ \begin{array}{rr}10 & -24 \-2 & 10 \-4 & 2\end{array} \right]\).

Key Concepts

Matrix Addition

When we talk about matrix addition, we are referring to the process of adding two matrices by adding their corresponding elements. To perform matrix addition, the matrices must be of the same dimension. This means that for matrix A (A = \left[\begin{array}{rr}3 & -6 \2 & -1 \-4 & 5\end{array}\right]), and matrix B (B = \left[\begin{array}{rr}1 & 0 \5 & -7 \-6 & 9\end{array}\right]), both are 3x2 matrices, which makes them compatible for addition.

• For the element in the first row and first column of the resulting matrix, we add the elements from A and B: \(3 + 1 = 4\).
• For the first row, second column, the result is \(-6 + 0 = -6\).
• Continue this process for every corresponding element.

Thus, the result of adding matrix A and B is: \[A + B = \left[\begin{array}{rr}4 & -6 \7 & -8 \-10 & 14\end{array}\right]\]. Matrix addition is a simple operation once you match each element correctly.

Matrix Subtraction

Matrix subtraction, like addition, involves handling each pair of corresponding elements from two matrices. The requirement for the matrices to be of the same size holds here as well. In matrix subtraction, each element in the resulting matrix is obtained by subtracting the element of the second matrix from the element of the first. Let's consider matrices A and B again:

• For the element in the first row and first column: \(3 - 1 = 2\).
• The element in the first row, second column becomes \(-6 - 0 = -6\).
• Proceed in this manner for the remaining elements.

The result of subtracting matrix B from matrix A is: \[A - B = \left[\begin{array}{rr}2 & -6 \-3 & 6 \2 & -4\end{array}\right]\]. This operation helps when determining differences between corresponding positions in two sets of data represented as matrices.

Scalar Multiplication

Scalar multiplication involves multiplying every element of a matrix by a scalar (a constant). This operation is straightforward and requires no change to the matrix's dimensions. For example, if you multiply matrix A by 2, you multiply every element in A by 2. The operation can be visualized as:

• First element: \(3 \times 2 = 6\).
• Next: \(-6 \times 2 = -12\).
• Continue this way through all elements.
• This results in a new matrix: \(2A = \left[\begin{array}{rr}6 & -12 \4 & -2 \-8 & 10\end{array}\right]\).

Similarly, multiplying matrix B by 3 results in another resulting matrix, which is used for further operations like addition or subtraction. This concept illustrates how matrices can be scaled, thereby increasing or decreasing their magnitude uniformly across all entries.

Matrix Arithmetic

Matrix arithmetic encompasses a variety of operations that can be performed on matrices including addition, subtraction, and scaling followed by addition or subtraction. Understanding these operations is crucial for disciplines such as computer graphics, physics, and statistics. Let's explore how they come together in combined operations like \(2A + 3B\) and \(4A - 2B\).

• In \(2A + 3B\), we first perform scalar multiplication on matrices A and B: \(2A\) and \(3B\). Then, their results are added, combining corresponding elements.
• Meanwhile, \(4A - 2B\) uses scalar multiplication followed by subtraction. After obtaining \(4A\) and \(2B\), you subtract elements from the latter from the former.

The calculations turn these matrices into solutions: \[2A + 3B = \left[\begin{array}{rr}9 & -12 \19 & -23 \-26 & 37\end{array}\right]\] and \[4A - 2B = \left[\begin{array}{rr}10 & -24 \-2 & 10 \-4 & 2\end{array}\right]\]. By understanding and practicing these operations, you build a strong foundation in matrix arithmetic essential for more advanced math and science studies.