Problem 6 - Precalculus

Question

For each pair of matrices \(A\) and \(B\) find the following, if defined \- \(3 A\) \- \(-B\) \- \(A^{2}\) \- \(A-2 B\) \- \(A B\) \- \(B A\) $$ A=\left[\begin{array}{rr} 1 & -2 \\ -3 & 4 \\ 5 & -6 \end{array}\right], B=\left[\begin{array}{lll} -5 & 1 & 8 \end{array}\right] $$

Step-by-Step Solution

Answer

Defined results are: \(3A\), \(-B\), \(BA\); undefined results are: \(A^2\), \(A-2B\), \(AB\).

Step 1: Calculate 3A

First, we need to multiply each element of matrix \( A \) by 3. Matrix \( A \) is 3x2, so:\[3A = 3 \times \begin{bmatrix} 1 & -2 \ -3 & 4 \ 5 & -6 \end{bmatrix} = \begin{bmatrix} 3 \times 1 & 3 \times -2 \ 3 \times -3 & 3 \times 4 \ 3 \times 5 & 3 \times -6 \end{bmatrix} = \begin{bmatrix} 3 & -6 \ -9 & 12 \ 15 & -18 \end{bmatrix}\]

Step 2: Calculate -B

Matrix \( B \) is a 1x3 matrix. To find \(-B\), multiply each element by -1.\[-B = -1 \times \begin{bmatrix} -5 & 1 & 8 \end{bmatrix} = \begin{bmatrix} 5 & -1 & -8 \end{bmatrix}\]

Step 3: Determine if A^2 is Defined

Matrix \( A^2 \) means multiplication of matrix \( A \) by itself. For matrix multiplication to be defined, the number of columns of the first matrix must equal the number of rows of the second matrix. Since \( A \) is 3x2, and \( A \) is 2x3, multiplication \( A \times A \) is undefined.

Step 4: Calculate A - 2B

To subtract \( 2B \) from \( A \), matrices \( A \) and \( 2B \) must have the same dimensions. First, calculate \( 2B \):\[2B = 2 \times \begin{bmatrix} -5 & 1 & 8 \end{bmatrix} = \begin{bmatrix} -10 & 2 & 16 \end{bmatrix}\]However, \( A \) (3x2) and \( 2B \) (1x3) do not have the same dimensions, so \( A - 2B \) is undefined.

Step 5: Calculate AB

The multiplication \( AB \) is defined if the number of columns in \( A \) is equal to the number of rows in \( B \). Matrix \( A \) is 3x2 and \( B \) is 1x3, hence \( AB \) is undefined since the dimensions (2 and 1) do not match.

Step 6: Calculate BA

For \( BA \) to be defined, the number of columns in matrix \( B \) must equal the number of rows in \( A \). Since \( A \) is 3x2 and \( B \) is 1x3, multiplication \( BA \) is defined and results in a 1x2 matrix.Calculate \( BA \):\[BA = \begin{bmatrix} -5 & 1 & 8 \end{bmatrix} \times \begin{bmatrix} 1 & -2 \ -3 & 4 \ 5 & -6 \end{bmatrix} = \begin{bmatrix} (-5)(1) + (1)(-3) + (8)(5) & (-5)(-2) + (1)(4) + (8)(-6) \end{bmatrix} = \begin{bmatrix} 19 & -16 \end{bmatrix}\]

Key Concepts

Matrix Addition

Matrix addition involves adding corresponding elements from two matrices to create a new matrix. This operation is only possible when both matrices have identical dimensions. For example, if you have matrices that are both 2x2 or both 3x3, you can add them together.
However, if the matrices do not share the same dimensions, matrix addition is undefined. In our exercise, since matrices \(A\) and \(B\) have different dimensions (3x2 and 1x3 respectively), we cannot perform matrix addition on them.

• Both matrices must have the same number of rows and columns.
• Add each element in the first matrix with the corresponding element in the second matrix.

Remember that each position \((i,j)\) in the new matrix will simply be the sum of the elements from the same position in both matrices.

Matrix Multiplication

Matrix multiplication is slightly more complex than addition, as it relies on specific size requirements between the matrices. To multiply two matrices, the number of columns in the first matrix must match the number of rows in the second matrix.
For instance, if matrix \(A\) is 3x2, it can only be multiplied by another matrix that has 2 rows, such as a 2x3 matrix.

• The resulting matrix will have dimensions that match the number of rows of the first matrix and the number of columns of the second matrix.
• Compute the elements by multiplying the rows of the first matrix with the columns of the second matrix and summing the products.

In our case, attempts to perform \(A^2\) or \(AB\) result in undefined operations due to mismatched dimensions. Conversely, \(BA\) is valid, resulting in a 1x2 matrix after calculation.

Scalar Multiplication

Scalar multiplication involves multiplying each element of a matrix by a single number, known as a scalar. This operation is straightforward and does not depend on the dimensions of the matrix.
For example, multiplying matrix \(A\) by a scalar like 3 means every element in \(A\) is tripled.

• Multiply each component of the matrix individually by the scalar.
• The dimensions of the matrix remain unchanged.

In the exercise, the operation \(3A\) was performed, smartly multiplying each element of \(A\) by 3 successfully. Similarly, \(-B\) was computed by multiplying each element of \(B\) by -1.

Matrix Dimensions

Understanding matrix dimensions is crucial for performing any matrix operations. A matrix's dimensions refer to its number of rows and columns, given in the form 'row x column'.
Dimensions determine how you can add, subtract, or multiply matrices with one another.

• Matrices \(A\) and \(B\) must have the same dimensions for addition or subtraction.
• For multiplication, the number of columns in the first matrix must equal the number of rows in the second matrix.
• The resulting matrix of a multiplication will have dimensions formed by the rows of the first matrix and the columns of the second.

In our scenario, matrix \(A\) is a 3x2 matrix while matrix \(B\) is a 1x3 matrix, highlighting the challenges faced when aiming to apply each operation due to their differing dimensions.