Solving For Matrix M Matrix Equation Examples

Hey guys! Today, we're diving into the fascinating world of matrix algebra to solve for an unknown matrix, M. Matrix operations might seem daunting at first, but with a step-by-step approach and a clear understanding of the rules, you'll be solving these problems like a pro in no time. So, let's jump right in and tackle this matrix challenge together!

Problem a: 7M = 3 * [[-4, 5], [0, 1]] + [[-2, 6], [0, 4]]

Let's break down the first part of our challenge. We need to find the matrix M that satisfies the equation 7M = 3 * [[-4, 5], [0, 1]] + [[-2, 6], [0, 4]]. This looks a bit intimidating, but don't worry, we'll take it one step at a time. The key here is to remember the order of operations and how matrix addition and scalar multiplication work.

Step 1: Scalar Multiplication

First, we need to deal with the scalar multiplication. We have 3 * [[-4, 5], [0, 1]]. This means we multiply each element of the matrix by 3. So, let's do that:

3 * [[-4, 5], [0, 1]] = [[3 * -4, 3 * 5], [3 * 0, 3 * 1]] = [[-12, 15], [0, 3]]

Now our equation looks like this: 7M = [[-12, 15], [0, 3]] + [[-2, 6], [0, 4]].

Step 2: Matrix Addition

Next up is matrix addition. We're adding the two matrices on the right side of the equation. Remember, to add matrices, they need to have the same dimensions, and we simply add the corresponding elements:

[[-12, 15], [0, 3]] + [[-2, 6], [0, 4]] = [[-12 + -2, 15 + 6], [0 + 0, 3 + 4]] = [[-14, 21], [0, 7]]

Now our equation is even simpler: 7M = [[-14, 21], [0, 7]].

Step 3: Solving for M

We're almost there! To find M, we need to isolate it. Since M is being multiplied by 7, we need to divide both sides of the equation by 7. Remember, dividing a matrix by a scalar means dividing each element of the matrix by that scalar:

M = (1/7) * [[-14, 21], [0, 7]] = [[-14/7, 21/7], [0/7, 7/7]] = [[-2, 3], [0, 1]]

So, we've found our matrix M for part a! M = [[-2, 3], [0, 1]]. See? Not so scary after all!

Problem b: [[9, 1], [2, 6]] + M = [[3, -5], [-1, 8]]

Okay, let's tackle the second part of our matrix puzzle. This time, we have the equation [[9, 1], [2, 6]] + M = [[3, -5], [-1, 8]]. Our mission, should we choose to accept it (and we do!), is to find the matrix M that makes this equation true.

The core concept here is similar to solving algebraic equations with numbers: we need to isolate M on one side of the equation. Since we have a matrix being added to M, we'll use the inverse operation: matrix subtraction.

Step 1: Isolating M

To get M by itself, we need to subtract the matrix [[9, 1], [2, 6]] from both sides of the equation. This is like subtracting a number from both sides of a regular equation. The equation becomes:

M = [[3, -5], [-1, 8]] - [[9, 1], [2, 6]]

Step 2: Matrix Subtraction

Now we perform the matrix subtraction. Just like addition, we subtract corresponding elements:

M = [[3 - 9, -5 - 1], [-1 - 2, 8 - 6]] = [[-6, -6], [-3, 2]]

And there you have it! We've found the matrix M for part b: M = [[-6, -6], [-3, 2]]. We successfully navigated another matrix challenge!

Key Concepts in Matrix Operations

Before we wrap up, let's quickly recap the key concepts we used to solve these problems. Understanding these principles is crucial for mastering matrix algebra. These concepts are the building blocks for more complex matrix operations and applications.

1. Scalar Multiplication

Scalar multiplication involves multiplying a matrix by a constant (a scalar). This means multiplying every element in the matrix by that scalar. It's like scaling the entire matrix up or down. For example, if you have a matrix A and a scalar k, then kA is a new matrix where each element of A has been multiplied by k. Scalar multiplication is a fundamental operation and is used extensively in linear transformations and other matrix applications.

2. Matrix Addition and Subtraction

Matrices can be added or subtracted only if they have the same dimensions (i.e., the same number of rows and columns). The operation involves adding or subtracting corresponding elements. For instance, if you have two matrices A and B of the same size, then A + B is a new matrix where each element is the sum of the corresponding elements in A and B. Matrix addition and subtraction are crucial for combining linear transformations and solving systems of linear equations.

3. Isolating the Unknown Matrix

Just like in regular algebra, solving for an unknown matrix often involves isolating it on one side of the equation. This might require using inverse operations, such as adding or subtracting matrices from both sides, or multiplying or dividing by a scalar. The goal is to manipulate the equation until you have M = [some matrix expression], where M is the unknown matrix you're trying to find. This is a core skill in linear algebra and is used in various applications, including solving systems of equations and finding matrix inverses.

Why Matrix Operations Matter

You might be wondering,