Problem 9 - Algebra 2

Question

Solve each matrix equation. Check your answers. $$ 3\left[\begin{array}{rr}{2} & {0} \\ {-1} & {5}\end{array}\right]-2 X=\left[\begin{array}{rr}{-10} & {5} \\ {0} & {17}\end{array}\right] $$

Step-by-Step Solution

Answer

The solution matrix X is \[ X = \left[\begin{array}{rr}{-5} & {5} \ {1.5} & {4}\end{array}\right] \].

Step 1: Multiply the Matrix by the Scalar

Multiply the scalar 3 with each element of the given matrix to obtain a new matrix.

Step 2: Set Up the Matrix Equation

Set up the matrix equation by placing the product from Step 1 on one side and the given matrix on the other side, all equal to the matrix X multiplied by the scalar 2.

Step 3: Solve for the Matrix X

To find the matrix X, divide each element of the matrix obtained after subtracting the product from Step 1 from the given matrix on the right side by the scalar 2.

Step 4: Verify the Solution

Multiply the resulting X matrix by 2 and check if the equation holds true by adding it to the product from Step 1 to ensure it equals the matrix given on the right side of the original equation.

Key Concepts

Scalar Multiplication

Scalar multiplication is one of the fundamental operations in matrix algebra. In essence, it involves multiplying every entry of a matrix by a constant value called a 'scalar'. Understanding scalar multiplication is crucial since it's often a step in more complex matrix manipulations.

For instance, in the exercise, multiplying the scalar 3 by the matrix \[ \begin{array}{rr}{2} & {0} \ {-1} & {5}\end{array} \] results in a new matrix, where every element is three times greater than the corresponding element in the original matrix. In mathematical terms, if the original matrix is labeled as A and the scalar as c, the product is obtained by computing \( cA \). Real-world applications of scalar multiplication include stretching or shrinking geometrical figures, which are represented as matrices in computer graphics.

Matrix Algebra

Matrix algebra encompasses various operations that can be performed on matrices, including addition, subtraction, multiplication, and scalar multiplication. Understanding these operations is crucial when solving matrix equations because they are the building blocks of more advanced procedures, like finding the determinant or inverse of a matrix.

In our exercise, the subtraction of two matrices is exemplified. Subtracting matrix B from matrix A requires subtracting each element in B from the corresponding element in A, like so: \( A - B \). It’s important to remember that matrix addition and subtraction are only defined for matrices of the same size and shape. Incorrect application of these operations is a common source of mistakes.

Solving Matrices

When solving matrices, the goal is to find the value of an unknown matrix that satisfies a given equation. This process typically involves using the principles of matrix algebra to manipulate the equation until the unknown matrix is isolated.

In our exercise, the matrix equation is solved by isolating the matrix \( X \). This often requires adding or subtracting matrices and using scalar multiplication, as seen in the step-by-step solution. Solving matrix equations is akin to solving algebraic equations with variables, except the variables are replaced by matrices; hence understanding element-wise operations is essential. The ability to simplify and rearrange equations is a skill needed for higher mathematics and problem-solving in fields such as engineering, physics, and economics.

Verifying Matrix Solutions

Verifying the solution of a matrix equation is as essential as finding the solution itself. It confirms the correctness of our answer. This verification involves substituting the found solution back into the original equation to see if the equation holds true.

In the context of our exercise, after determining the matrix \( X \), we verify its accuracy by performing the scalar multiplication and matrix addition as per the original matrix equation. If the left-hand side and right-hand side of the equation are equivalent upon substitution, the solution is verified. This check helps prevent and identify potential errors made during calculation, ensuring that the solution is valid. Establishing this habit can be beneficial not only in mathematics but also in real-life situations where confirmation of results is necessary.