How to Know if Two Matrices Are Equal with Variables

How to KNOW if TWO MATRICES are EQUAL (with unknowns)

Hi everyone, and welcome to the Futuros Sobresalientes channel! Today, we’re diving into the subject of mathematics, particularly focusing on the concept of equal matrices. If you’re not subscribed yet, make sure to do so; it’s free! You can also join the channel or support me on Patreon. And if you need private lessons, check out our website linked in the description. Let’s get started!

What Are Equal Matrices?

Two matrices are considered equal if they have the same dimensions and their corresponding elements are identical. In formal terms, matrices A and B are equal if:

Both matrices have the same number of rows (m) and columns (n).
Each corresponding element is equal, meaning A_ij = B_ij for all valid i and j.

This concept might sound complex, but don’t worry! We’ll clarify it with examples and exercises that will help clear up any doubts.

Example 1: Determining Equal Matrices

Let’s start with our first exercise: What value should x take for these matrices to be equal?

Matrix A

Matrix B

Matrix C

A =

1	2
3	4
x	5

B =

1	x – 1
3	3
9	5

C =

1	2
3	4
6	5

First, we check the dimensions of matrix A:

Dimensions of A: 3 x 2 (3 rows and 2 columns)

For matrix B:

Dimensions of B: 2 x 3 (2 rows and 3 columns)

We can observe that matrices A and B are not equal since they do not share the same dimensions. However, regarding matrices A and C, they have the same dimensions (3 x 2), so they might be equal if the corresponding elements are the same. This will occur if x = 6 for all elements in both matrices A and C to coincide.

Example 2: Calculate x, y, and z for Equal Matrices

Now, let’s advance to our next problem where we need to calculate the values of x, y, and z for the following matrices to be equal:

Matrix A

Matrix B

A =

1	x + 3
y + 1	5

B =

1	4
2	5

Let’s start comparing the dimensions:

Matrix A: 2 x 2
Matrix B: 2 x 2

Since they have the same dimensions, we can proceed to solving the elements:

1 = x + 3 → x = 1 – 3 → x = -2
y + 1 = 5 → y = 5 – 1 → y = 4
2 must correspond to z, hence z = 0

We now know that:

x = -2
y = 4
z = 0

Final Thoughts

Today, we explored the fascinating world of matrices, focusing on how to determine if two matrices are equal, especially in the context of unknown variables. Remember that equal matrices must have the same dimensions, and each corresponding element must match.

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