Problem 9 - College Algebra

Question

Find the following matrices: a. \(A+B\) b. \(A-B\) c. \(-4 A\) d. \(3 A+2 B\) $$A=\left[\begin{array}{ll}4 & 1 \\\3 & 2\end{array}\right], \quad B=\left[\begin{array}{ll}5 & 9 \\\0 & 7\end{array}\right]$$

Step-by-Step Solution

Answer

a) \(A+B=\left[\begin{array}{ll}9 & 10 \3 & 9\end{array}\right]\), b) \(A-B=\left[\begin{array}{ll}-1 & -8 \3 & -5\end{array}\right]\), c) \(-4A=\left[\begin{array}{ll}-16 & -4\-12 & -8\end{array}\right]\), d) \(3A+2B=\left[\begin{array}{ll}22 & 21\9 & 20\end{array}\right]\)

Step 1: Addition of Matrices

Here, each element of matrix A is added to its corresponding one in matrix B, resulting in \[ C = A + B = \left[\begin{array}{ll}4+5 & 1+9 \3+0 & 2+7\end{array}\right] = \left[\begin{array}{ll}9 & 10 \3 & 9\end{array}\right] \]

Step 2: Subtraction of Matrices

In this step, each element in matrix B is subtracted from its corresponding one in matrix A. This results in \[ D = A - B = \left[\begin{array}{ll}4-5 & 1-9 \3-0 & 2-7\end{array}\right] = \left[\begin{array}{ll}-1 & -8 \3 & -5\end{array}\right] \]

Step 3: Multiplication of a Matrix by a Scalar

First multiply each element of matrix A by -4, resulting in: \[ E = -4 \cdot A = \left[\begin{array}{ll}-4 \cdot 4 & -4 \cdot 1\-4 \cdot 3 & -4 \cdot 2\end{array}\right] = \left[\begin{array}{ll}-16 & -4\-12 & -8\end{array}\right] \]

Step 4: Combination of Matrix Addition and Scalar Multiplication

First multiply each element in matrix A by 3 and each element in matrix B by 2. Then add the corresponding elements, resulting in \[ F = 3A + 2B = \left[\begin{array}{ll}3 \cdot 4+2 \cdot 5 & 3 \cdot 1+2 \cdot 9\3 \cdot 3+2 \cdot 0 & 3 \cdot 2+2 \cdot 7\end{array}\right] = \left[\begin{array}{ll}22 & 21\9 & 20\end{array}\right] \]

Key Concepts

Matrix Addition

When we talk about adding two matrices, just imagine you're throwing a party and bringing together guests from two different groups. Each guest from the first group (Matrix A) pairs up with a corresponding guest from the second group (Matrix B), and they exchange numbers. In matrix addition, these numbers are simply added together.

For example, if Matrix A has a '4' and Matrix B brings a '5' to the party in the same position, the sum is '9'. This process is repeated for each position in the matrices, creating a new matrix where all the positions are the combined values from the original pairs.

You can only add matrices together if they are of the same size. Try to remember, if the matrices are friends of different sizes, they can't merge their parties into one.

Matrix Subtraction

If matrix addition is like merging parties, matrix subtraction is like planning a dance battle where each participant from Matrix A competes against their counterpart in Matrix B. Instead of adding their moves together, you subtract the moves of the second group from the first.

In our textbook example, if one dancer from Matrix A has '4' moves, but their rival from Matrix B has '5', the result is '-1'. This negative score shows that the dancer from Matrix B outperformed their counterpart. Continue this dance-off across the entire dance floor (each position in the matrices) to get your final dance battle scorecard (the result of the matrix subtraction).

Remember, for this dance-off to happen, the matrices must have the same size - unequal groups can't compete fairly!

Scalar Multiplication of Matrices

Imagine each matrix as a cupcake recipe, and the scalar as the number of batches you're making. If you have to multiply a matrix (the recipe) by a scalar (let's say '-4' batches), you're adjusting each ingredient amount by that scalar.

In our case, multiplying each element of Matrix A by '-4' means we're making four times as many cupcakes, but with a twist – since the scalar is negative, it's like we're taking away cupcakes instead of adding them to the party. Each ingredient (each value in Matrix A) gets multiplied by '-4', and thus, we end up with a completely new set of values (Matrix E) that tell us how many cupcakes we're left with (or owe) for the bake sale.

Combining Matrix Operations

When you combine matrix operations like addition and scalar multiplication, it's like you're hosting a cooking show and preparing multiple recipes at once. You're multitasking by first adjusting your original recipes (matrices) according to the number of guests (scalar multiplication) and then blending two different dishes together (matrix addition) to create a fusion that has the best of both worlds.

The final dish (resultant matrix) is a sumptuous combination of flavors, with the quantities of ingredients carefully adjusted and mixed together. The steps you take ensure that the proportions are just right, combining the innate structure of matrices with the stretch-and-shrink effect of scalars. Just make sure to keep track of the order in which you mix your ingredients to keep your guests happy and coming back for more!