Solving Matrix Equations

Just like linear equations and quadratic equations, we may be asked to solve matrix equations. Although this might seem daunting at first, we will soon learn a few methods that can make this process relatively straightforward. Let's find out more:

What is a matrix equation?

A matrix equation is just like a normal equation. The only real difference is that we have swapped out our variables for numbers (we call these elements when we place them in a matrix). Consider the following system:

Note that this example is easy because all of our variables are lined up in the same order. Our "constraints" are also on the right side. If our system doesn't look like this, we might need to rearrange it a little before we move on. Each coefficient becomes an element on the resulting matrix:

Adding matrices

When we add matrices, all we need to do is add the numbers in the matching positions. For example:

Here we would add:

Note that the same basic principles apply to subtracting matrices.

Multiplying matrices by a scalar

Multiplying matrices is also pretty easy. All we need to do is multiply the matrix by a constant.

For example,

Note that this is called scalar multiplication.

Examples of solving matrices using addition and scalar multiplication.

Consider the following matrix:

Can we solve this matrix?

So,

Now consider this matrix:

Can we solve this one?

Hence,

Solving a system of linear equations using matrices

We can also use matrix equations to solve systems of linear equations. We can do this by working with the left and right sides of our equations:

Let's say we turned this system of linear equations:

Into this matrix:

Let's focus on the coefficient matrix (the left-hand matrix) and try to find the inverse:

Now we can multiply each side of the matrix equation by the inverse matrix. Remember that matrix multiplication is not commutative, so we need to place our inverse matrix on the left of each side of the equation:

The identity matrix on the left shows us that we calculated our inverse matrix correctly.

Now we can eliminate this value, leaving us with:

Therefore, our solution is $\left(4,-5\right)$ .

Topics related to the Solving Matrix Equations

Distributive Property of Matrices

Adding and Subtracting Matrices

Square Matrix

Flashcards covering the Solving Matrix Equations

Linear Algebra Flashcards

Numerical Methods Flashcards

Practice tests covering the Solving Matrix Equations

Linear Algebra Diagnostic Tests

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