Problem 12 - Precalculus with Limits

Question

In Exercises 11-18, if possible, find (a) \(A+B\), (b) \(A-B\), (c ) \(3A\), and (d) \(3A-2B\). \(A = \left[ \begin{array}{r} 8 & -1 \\ 2 & 3 \\ -4 & 5 \end{array} \right]\), \(B = \left[ \begin{array}{r} 1 & 6 \\ -1 & -5 \\ 1 & 10 \end{array} \right]\)

Step-by-Step Solution

Answer

(a) \(A+B = \left[ \begin{array}{r} 9 & 5 \ 1 & -2 \ -3 & 15 \end{array} \right]\) (b) \(A-B = \left[ \begin{array}{r} 7 & -7 \ 3 & 8 \ -5 & -5 \end{array} \right]\) (c ) \(3A = \left[ \begin{array}{r} 24 & -3 \ 6 & 9 \ -12 & 15 \end{array} \right]\) (d) \(3A-2B = \left[ \begin{array}{r} 22 & -15 \ 8 & 19 \ -14 & -5 \end{array} \right]\)

Step 1: Adding Matrices

Two matrices can be added together by adding the corresponding entries in each matrix. In this case, that gives \(A + B = \left[ \begin{array}{r} 8+1 & -1+6\ 2+(-1) & 3+(-5)\ -4+1 & 5+10 \end{array} \right]\), which simplifies to \( \left[ \begin{array}{r} 9 & 5 \ 1 & -2 \ -3 & 15 \end{array} \right]\).

Step 2: Subtracting Matrices

Matrix subtraction is done similarly to addition: subtract each corresponding entry in the matrices. So, \(A - B = \left[ \begin{array}{r} 8-1 & -1-6\ 2-(-1) & 3-(-5)\ -4-1 & 5-10 \end{array} \right]\), which simplifies to \( \left[ \begin{array}{r} 7 & -7 \ 3 & 8 \ -5 & -5 \end{array} \right]\).

Step 3: Scalar Multiplication

Scalar multiplication requires each entry of the matrix to be multiplied by the scalar. So, \( 3A = 3 * \left[ \begin{array}{r} 8 & -1\ 2 & 3\ -4 & 5 \end{array} \right]\), which becomes \( \left[ \begin{array}{r} 24 & -3 \ 6 & 9 \ -12 & 15 \end{array} \right]\).

Step 4: Combine Operations

Finally, for \(3A - 2B\), first each matrix is multiplied by its scalar: \(3A= \left[ \begin{array}{r} 24 & -3 \ 6 & 9 \ -12 & 15 \end{array} \right]\), and \(2B= 2*\left[ \begin{array}{r} 1 & 6\ -1 & -5\ 1 & 10 \end{array} \right]\), resulting in \(2B=\left[ \begin{array}{r} 2 & 12\ -2 & -10\ 2 & 20 \end{array} \right]\) . Then subtract the corresponding entries in each resulting matrix: \(3A - 2B = \left[ \begin{array}{r} 24-2 & -3-12\ 6-(-2) & 9-(-10)\ -12-2 & 15-20 \end{array} \right]\), simplifying to \( \left[ \begin{array}{r} 22 & -15 \ 8 & 19 \ -14 & -5 \end{array} \right]\).

Key Concepts

Adding Matrices

When it comes to matrix operations, one of the most fundamental is matrix addition. To add two matrices together, they must be of the same dimensions, that is, they must have the same number of rows and columns. The process involves adding the corresponding entries—those in the same row and column position—from each matrix.

For example, if we have two matrices, A and B, where,
\(A = \left[ \begin{array}{r} 8 & -1 \ 2 & 3 \ -4 & 5 \end{array} \right]\) and
\(B = \left[ \begin{array}{r} 1 & 6 \ -1 & -5 \ 1 & 10 \end{array} \right]\),

the sum \(A + B\) would be calculated as follows:

• For the entry in the first row, first column: \(8 + 1 = 9\)
• For the entry in the first row, second column: \(-1 + 6 = 5\)
• This process is repeated for each corresponding entry.

The result is a new matrix with the same dimensions as matrices A and B, containing the sums of the corresponding entries of these matrices.

Subtracting Matrices

Subtracting matrices is another core operation in matrix arithmetic. It's similar to addition in that we must have two matrices of identical dimensions. To subtract matrix B from matrix A, we simply subtract the corresponding entries of B from A.

Using our matrices A and B from the previous example, the difference \(A - B\) is found by taking each entry of A and subtracting the corresponding entry of B. For instance:

• For the entry in the first row, first column: \(8 - 1 = 7\)
• For the entry in the first row, second column: \(-1 - 6 = -7\)
• The process is replicated across all entries.

It results in a new matrix where each entry reflects the difference between the entries of matrices A and B.

Scalar Multiplication of Matrices

Scalar multiplication is a type of matrix operation where every entry in a matrix is multiplied by a fixed number, known as a scalar. This operation affects the scale of the matrix but doesn't change its dimensions.

For instance, if we multiply matrix A by scalar 3, indicated as \(3A\), every entry of A is multiplied by 3:

• The entry in the first row and first column: \(3 \times 8 = 24\)
• The entry in the first row and second column: \(3 \times -1 = -3\)
• This multiplication carries through to every entry.

Scalar multiplication results in each entry of the original matrix being amplified by the scalar value to form a new matrix.

Matrix Arithmetic

Matrix arithmetic encompasses operations like addition, subtraction, and scalar multiplication. These operations can be combined to perform more complex calculations. For instance, to calculate an expression like \(3A - 2B\), you must first perform scalar multiplication on each matrix and then subtract one from the other.

Initially, we multiply matrices A and B by scalars 3 and 2 respectively and then subtract the resulting matrix of \(2B\) from \(3A\). Performing these steps entails:

• Applying scalar multiplication to get \(3A\) and \(2B\).
• Then, subtracting \(2B\) from \(3A\) by taking each corresponding entry in \(3A\) and subtracting the corresponding entry in \(2B\).

The outcome of such matrix arithmetic is a new matrix that combines the results of various operations based on the rules of matrix addition, subtraction, and scalar multiplication.