Problem 24 - Precalculus with Limits

Question

In Exercises 19-24, evaluate the expression. \(-\left[\begin{array}{r} 4 & 11 \\ -2 & -1 \\ 9 & 3 \end{array}\right] + \dfrac{1}{6}\left( \left[\begin{array}{r} -5 & -1 \\ 3 & 4 \\ 0 & 13 \end{array}\right] + \left[\begin{array}{r} 7 & 5 \\ -9 & -1 \\ 6 & -1 \end{array}\right]\right)\)

Step-by-Step Solution

Answer

The result is \[\begin{array}{r} -\dfrac{13}{3} & -\dfrac{35}{3} \\ 3 & \dfrac{1}{2} \\ -10 & -5\end{array}\]

Step 1: Matrix Addition and Subtraction

Combine the two matrices within the parenthesis:\n \[\begin{array}{r} -5 & -1 \\ 3 & 4 \\ 0 & 13 \end{array} + \begin{array}{r} 7 & 5 \\ -9 & -1 \\ 6 & -1 \end{array} = \begin{array}{r} (-5+7) & (-1+5) \\ (3-9) & (4-1) \\ (0+6) & (13-1) \end{array} = \begin{array}{r} 2 & 4 \\ -6 & 3 \\ 6 & 12\end{array}\]

Step 2: Scalar Multiplication

Next, perform scalar multiplication by multiplying each element in the matrix obtained in step 1 by 1/6.\n\[\dfrac{1}{6}\begin{array}{r} 2 & 4\\ -6 & 3\\ 6 & 12\end{array} = \begin{array}{r} \dfrac{1}{6} * 2 & \dfrac{1}{6} * 4\\ \dfrac{1}{6} * -6 & \dfrac{1}{6} * 3\\ \dfrac{1}{6} * 6 & \dfrac{1}{6} * 12\end{array} = \begin{array}{r} \dfrac{1}{3} & \dfrac{2}{3}\\ -1 & \dfrac{1}{2}\\ 1 & 2\end{array}\]

Step 3: Final Addition

Subtract the original matrix from the matrix obtained in step 2. \[ - \begin{array}{r} 4 & 11\\ -2 & -1\\ 9 & 3\end{array} + \begin{array}{r} \dfrac{1}{3} & \dfrac{2}{3}\\ -1 & \dfrac{1}{2}\\ 1 & 2\end{array} = \begin{array}{r} -4 - \dfrac{1}{3} & -11 - \dfrac{2}{3}\\ 2 - (-1) & 1 - \dfrac{1}{2}\\ -9 - 1 & -3 - 2\end{array} = \begin{array}{r} -\dfrac{13}{3} & -\dfrac{35}{3}\\ 3 & \dfrac{1}{2}\\ -10 & -5\end{array}\]

Key Concepts

Matrix Addition

Matrix addition is a fundamental operation when dealing with matrices. To add two matrices, they must have the same dimensions, which means each matrix should have the same number of rows and the same number of columns. The way it works is pretty simple: you add corresponding elements from each matrix together.

For example, if you have two matrices:

• Matrix A: \( \begin{array}{cc} a & b \ c & d \end{array} \)
• Matrix B: \( \begin{array}{cc} e & f \ g & h \end{array} \)

The addition of these matrices (\( A + B \)) results in a new matrix where each element is the sum of the corresponding elements:
\[ \begin{array}{cc} a+e & b+f \ c+g & d+h \end{array} \]
This operation is useful in many areas such as solving systems of equations and in various applications in physics and engineering.

Scalar Multiplication

Scalar multiplication is another important operation in matrix algebra. It involves multiplying each element of a matrix by a single number, called a scalar. This operation is very straightforward, but it's crucial in scaling the matrix's size or transforming its values uniformly.

Given a matrix \( M = \begin{array}{cc} x & y \ z & w \end{array} \) and a scalar \( k \), the result of the scalar multiplication \( kM \) will be a new matrix:

• \( \begin{array}{cc} kx & ky \ kz & kw \end{array} \)

For instance, if you have a scalar \( \dfrac{1}{6} \) and apply it to a matrix:

• \( \dfrac{1}{6} \begin{array}{cc} 2 & 4 \ -6 & 3 \end{array} \)

The resulting matrix will be:

• \( \begin{array}{cc} \dfrac{2}{6} & \dfrac{4}{6} \ \dfrac{-6}{6} & \dfrac{3}{6} \end{array} = \begin{array}{cc} \dfrac{1}{3} & \dfrac{2}{3} \ -1 & \dfrac{1}{2} \end{array} \)

This procedure is essential for various matrix transformations and is applied in many practical scenarios, such as graphics processing and solving linear algebra problems.

Matrix Subtraction

Matrix subtraction follows a similar procedure to matrix addition. Like addition, both matrices involved in a subtraction operation must be of the same size. The subtraction is performed by deducting corresponding elements of the matrices.

For example, consider two matrices:

• Matrix A: \( \begin{array}{cc} m & n \ o & p \end{array} \)
• Matrix B: \( \begin{array}{cc} q & r \ s & t \end{array} \)

The result of their subtraction (\( A - B \)) will be:\[ \begin{array}{cc} m-q & n-r \ o-s & p-t \end{array} \]
In the provided exercise, you will see that matrix subtraction is used after the additions and scalar multiplications have been completed. It's used frequently in various fields, like computer science and statistical computing, to find differences in datasets or decrement values matrix-wise.