Problem 2 - Precalculus

Question

For each pair of matrices \(A\) and \(B\) find the following, if defined \- \(3 A\) \- \(-B\) \- \(A^{2}\) \- \(A-2 B\) \- \(A B\) \- \(B A\) $$ A=\left[\begin{array}{ll} -1 & 5 \\ -3 & 6 \end{array}\right], B=\left[\begin{array}{rr} 2 & 10 \\ -7 & 1 \end{array}\right] $$

Step-by-Step Solution

Answer

3A = \([-3, 15; -9, 18]\), -B = \([-2, -10; 7, -1]\), A² = \([-14, 25; -15, 21]\), A - 2B = \([-5, -15; 11, 4]\), AB = \([-37, -5; -48, -24]\), BA = \([-32, 70; 4, -29]\).

Step 1: Calculate 3A

To find \(3A\), multiply each element of matrix \(A\) by 3:\[3A = 3 \cdot \begin{bmatrix}-1 & 5 \ -3 & 6\end{bmatrix} = \begin{bmatrix}3(-1) & 3(5) \ 3(-3) & 3(6)\end{bmatrix} = \begin{bmatrix}-3 & 15 \ -9 & 18\end{bmatrix}\]

Step 2: Calculate -B

To find \(-B\), multiply each element of matrix \(B\) by -1:\[-B = -1 \cdot \begin{bmatrix}2 & 10 \ -7 & 1\end{bmatrix} = \begin{bmatrix}-2 & -10 \ 7 & -1\end{bmatrix}\]

Step 3: Calculate A²

To find \(A^2\), multiply matrix \(A\) by itself. Using matrix multiplication rules, calculate as follows:\[A^2 = \begin{bmatrix}-1 & 5 \ -3 & 6\end{bmatrix} \cdot \begin{bmatrix}-1 & 5 \ -3 & 6\end{bmatrix}\]Calculate the products:- First row, first column: \((-1)(-1) + (5)(-3) = 1 - 15 = -14\)- First row, second column: \((-1)(5) + (5)(6) = -5 + 30 = 25\)- Second row, first column: \((-3)(-1) + (6)(-3) = 3 - 18 = -15\)- Second row, second column: \((-3)(5) + (6)(6) = -15 + 36 = 21\)Thus,\[A^2 = \begin{bmatrix}-14 & 25 \ -15 & 21\end{bmatrix}\]

Step 4: Calculate A-2B

To find \(A - 2B\), first calculate \(2B\) and then subtract it from \(A\):\[2B = 2 \cdot \begin{bmatrix}2 & 10 \ -7 & 1\end{bmatrix} = \begin{bmatrix}4 & 20 \ -14 & 2\end{bmatrix}\]Now subtract \(2B\) from \(A\):\[A - 2B = \begin{bmatrix}-1 & 5 \ -3 & 6\end{bmatrix} - \begin{bmatrix}4 & 20 \ -14 & 2\end{bmatrix} = \begin{bmatrix}-1-4 & 5-20 \ -3+14 & 6-2\end{bmatrix} = \begin{bmatrix}-5 & -15 \ 11 & 4\end{bmatrix}\]

Step 5: Calculate AB

To find \(AB\), multiply matrix \(A\) by matrix \(B\):\[AB = \begin{bmatrix}-1 & 5 \ -3 & 6\end{bmatrix} \cdot \begin{bmatrix}2 & 10 \ -7 & 1\end{bmatrix}\]Calculate the products:- First row, first column: \((-1)(2) + (5)(-7) = -2 - 35 = -37\)- First row, second column: \((-1)(10) + (5)(1) = -10 + 5 = -5\)- Second row, first column: \((-3)(2) + (6)(-7) = -6 - 42 = -48\)- Second row, second column: \((-3)(10) + (6)(1) = -30 + 6 = -24\)Thus, \[AB = \begin{bmatrix}-37 & -5 \ -48 & -24\end{bmatrix}\]

Step 6: Calculate BA

To find \(BA\), multiply matrix \(B\) by matrix \(A\):\[BA = \begin{bmatrix}2 & 10 \ -7 & 1\end{bmatrix} \cdot \begin{bmatrix}-1 & 5 \ -3 & 6\end{bmatrix}\]Calculate the products:- First row, first column: \((2)(-1) + (10)(-3) = -2 - 30 = -32\)- First row, second column: \((2)(5) + (10)(6) = 10 + 60 = 70\)- Second row, first column: \((-7)(-1) + (1)(-3) = 7 - 3 = 4\)- Second row, second column: \((-7)(5) + (1)(6) = -35 + 6 = -29\)Thus, \[BA = \begin{bmatrix}-32 & 70 \ 4 & -29\end{bmatrix}\]

Key Concepts

Matrix Multiplication

Matrix multiplication involves a unique way to combine two matrices to form a new matrix. Unlike number multiplication, matrix multiplication is not element-wise; instead, it involves multiplying rows by columns.
To multiply two matrices, the number of columns in the first matrix must equal the number of rows in the second. For example, if you're multiplying two matrices, say \(A\) (an \(m \times n\) matrix) and \(B\) (an \(n \times p\) matrix), the resulting matrix will have dimensions \(m \times p\).
The process is done by taking each row from the first matrix and pairing it with each column of the second matrix. The individual elements are multiplied and summed up to form an element of the result matrix.

• Example: To compute a particular element, like the first row, first column of the resulting matrix, we'll multiply the first row of the first matrix by the first column of the second and sum the products.
• It’s important to note that matrix multiplication is not commutative. This means that \(AB\) is not necessarily equal to \(BA\).

Scalar Multiplication

Scalar multiplication is much simpler than matrix multiplication and involves multiplying every element of a matrix by the same numerical value, known as a scalar. This operation scales the matrix without changing its structure or size.
Consider a matrix \(C\) and a scalar \(k\). To perform scalar multiplication, simply multiply every element in \(C\) by \(k\).

• Example: If \(C = \begin{bmatrix}3 & 1 \ 2 & 4\end{bmatrix}\) and the scalar is 3, the result is: \[ 3C = \begin{bmatrix}3(3) & 3(1) \ 3(2) & 3(4)\end{bmatrix} = \begin{bmatrix}9 & 3 \ 6 & 12\end{bmatrix} \]

Scalar multiplication is an essential operation in linear algebra, used in scaling transformations and other matrix manipulations.

Matrix Subtraction

Matrix subtraction is the process of subtracting one matrix from another, provided that they are of the same dimensions. Like matrix addition, matrix subtraction is performed element-wise, meaning each corresponding element of the two matrices is subtracted one by one.
To subtract matrix \(B\) from matrix \(A\), described as \(A - B\), you need to ensure both matrices have the same shape. If \(A\) and \(B\) are both \(m \times n\) matrices, then:

• The element in the first row, first column of the result is the element in the first row, first column of \(A\) minus the element in the first row, first column of \(B\).
• This operation applies across all corresponding elements.

For example, subtracting matrix \(B = \begin{bmatrix}2 & 3 \ 5 & 6\end{bmatrix}\) from matrix \(A = \begin{bmatrix}5 & 7 \ 8 & 10\end{bmatrix}\) yields:\[A - B = \begin{bmatrix}5-2 & 7-3 \ 8-5 & 10-6\end{bmatrix} = \begin{bmatrix}3 & 4 \ 3 & 4\end{bmatrix} \]

Matrix Addition

Matrix addition is one of the simplest operations in matrics manipulation. It involves adding two matrices together by summing up their corresponding elements. Just like subtraction, both matrices need to have the same dimensions.
If matrix \(A\) and matrix \(B\) are both \(m \times n\) matrices, their sum, \(A + B\), is obtained by adding each element of \(A\) with the corresponding element of \(B\):

• For the element in the first row and first column of the resulting matrix, you add the first row, first column of \(A\) to the first row, first column of \(B\).
• This process continues for all elements.

Example: Adding matrices \(D = \begin{bmatrix}1 & 2 \ 3 & 4\end{bmatrix}\) and \(E = \begin{bmatrix}4 & 3 \ 2 & 1\end{bmatrix}\) results in:\[D + E = \begin{bmatrix}1+4 & 2+3 \ 3+2 & 4+1\end{bmatrix} = \begin{bmatrix}5 & 5 \ 5 & 5\end{bmatrix} \]Matrix addition is used extensively in applications such as computer graphics and solving systems of equations.