Problem 59 - Algebra 2

Question

Solve \(3 Y+2\left[\begin{array}{rr}{-1} & {-3} \\ {2} & {5}\end{array}\right]=\left[\begin{array}{cc}{13} & {-9} \\ {4} & {16}\end{array}\right] .\) Show the steps of your solution.

Step-by-Step Solution

Answer

After solving the equation, the matrix Y is equal to \[Y = \left[\begin{array}{rr}{3} & {-1} \ {0} & {3}\end{array}\right]\].

Step 1: Distribute the Scalar

Distribute the scalar value 2 across the elements of the 2x2 matrix. This step involves multiplication of each element of the matrix by 2.

Step 2: Write the Multiplied Matrix

After distributing the scalar value, the resulting matrix is obtained. Replace the multiplication with the resulting matrix.

Step 3: Isolate the Variable Y

The variable Y is being multiplied by 3 and is a matrix. To isolate Y, you need to divide each element of the resulting equation by 3.

Step 4: Solve for the Matrix Y

Divide each element of the result from Step 3 by 3 to get the values of Y.

Key Concepts

Matrix Multiplication

Matrix multiplication is a fundamental operation in linear algebra that combines two matrices to produce a third matrix. However, it's different from simple element-wise multiplication and requires a specific process to get right.

Here’s how it generally works:

• Imagine you have two matrices, A and B, where A is of size m x n and B is n x p.
• You can multiply A by B only if the number of columns in A matches the number of rows in B (n).
• The resultant matrix will be of size m x p, where each element is the sum of the products of the respective elements from the rows of A and the columns of B.

When solving matrix equations, getting the multiplication right is crucial because it affects the entire calculation that follows.

Scalar Multiplication

Scalar multiplication, another essential operation in the matrix toolbox, involves multiplying every element of a matrix by a single number, referred to as a scalar. It's much simpler than matrix multiplication and works as follows:

• Let's say you have a matrix A and a scalar k.
• To perform scalar multiplication, you multiply every single element of A by k, which gives you a new matrix where every element is k times the original element in A.

This is a vital step when solving systems of equations involving matrices, as it allows for combinations and modifications of equations without altering their solutions.

Isolating Variables in Matrices

Finally, isolating a variable in a matrix equation is parallel to isolating variables in traditional algebra.

The goal is to 'solve for' a matrix, meaning to find its value in the simplest form. Just like with numbers, you might add, subtract, multiply, or divide to do this with matrices.

• To isolate a matrix variable, you employ inverse operations that affect the entire matrix.
• If a matrix variable is being multiplied by a scalar, as in our exercise, you would divide every element of the matrix by that scalar.
• If it's being added or subtracted, you'd do the respective inverse operation to both sides of the equation.

Through these manipulations, you can successfully 'solve' for the matrix variable.