Problem 20 - Algebra 2

Question

Perform the indicated matrix operations. If the matrix does not exist, write impossible. $$ \left[\begin{array}{r}{3} \\ {-8} \\\ {-2}\end{array}\right]-\left[\begin{array}{r}{-4} \\ {5} \\\ {-2}\end{array}\right] $$

Step-by-Step Solution

Answer

The resulting matrix is \( \left[\begin{array}{r} {7} \\ {-13} \\ {0} \end{array}\right] \).

Step 1: Understand the Operation

The operation to perform is matrix subtraction. We have two matrices, each with one column and three rows. The subtraction of two matrices of the same size means simply subtracting the corresponding elements.

Step 2: Subtract Corresponding Elements

To subtract the matrices, subtract each corresponding element from the two matrices: \[\left[\begin{array}{r} {3} \ {-8} \ {-2} \end{array}\right] - \left[\begin{array}{r} {-4} \ {5} \ {-2} \end{array}\right] = \left[\begin{array}{r} {3 - (-4)} \ {-8 - 5} \ {-2 - (-2)} \end{array}\right]\]

Step 3: Calculate Each Element

Evaluate each of the element-wise subtractions: - The first element: \(3 - (-4) = 3 + 4 = 7\)- The second element: \(-8 - 5 = -8 - 5 = -13\)- The third element: \(-2 - (-2) = -2 + 2 = 0\)Thus, the resulting matrix is: \[\left[\begin{array}{r} {7} \ {-13} \ {0} \end{array}\right]\]

Step 4: Check the Result

Verify that the subtraction operation has been completed correctly by ensuring each element has been correctly deducted.

Key Concepts

Element-wise Subtraction

Matrix subtraction is performed by subtracting each corresponding element from one matrix with the other. Both matrices need to be of the same size to do this. This process is known as element-wise subtraction. It involves subtracting individual elements that are located in the same position within the matrices.

• Think of it as doing separate small subtraction operations for each pair of corresponding elements.
• No element should be left out during the subtraction.
• Pay attention to signs and operation order; subtracting a negative number is equivalent to adding its absolute value.

For example, in our exercise, the first matrix element in the top row is 3 and the corresponding element in the second matrix is -4. The operation becomes: \[3 - (-4) = 3 + 4 = 7\] By going through the matrix, element by element, you ensure each one is properly subtracted, leading to the correct result.

Matrix Operations

Matrix operations include a variety of manipulations such as addition, subtraction, multiplication, and finding inverses or determinants. Subtraction, being one of the basic operations, requires two matrices to be of the same dimensions.

• Always ensure matrix alignment; both matrices should have the same number of rows and columns.
• Matrix subtraction is not commutative; the order of matrices matters.
• Check that each operation is valid before proceeding.

When performing matrix subtraction, like in our exercise:\[\left[\begin{array}{r}3 \ -8 \ -2\end{array}\right] - \left[\begin{array}{r}-4 \ 5 \ -2\end{array}\right]\]make sure each step of element-wise subtraction is double-checked for errors.

Matrices

Matrices are mathematical structures consisting of rows and columns filled with numbers, symbols, or expressions. They are used in a wide range of mathematical applications from solving equations to representing complex data sets.

• A single element in a matrix is called an entry.
• The dimension of a matrix is given by the number of rows and columns it contains.
• Matrices emerge in various fields, enabling data transformation, manipulation, and representation.

In our exercise, both matrices are vectors with three elements each, arranged in a single column:\[\left[\begin{array}{c}3 \ -8 \ -2\end{array}\right]\] Understanding the dimensions and configuration of matrices is crucial when attempting operations like addition or subtraction. This ensures that each matrix operation is appropriate and the results are accurate.