# Matrices Properties

## Matrices Properties - How it Works - Video

### How to Label Matrices

How to Label Matrices:

Each matrix is labeled rows x columns, so there isn't any confusion. You count how many horizontal lines and then count how many vertical lines there are. Hopefully they are the same then it doesn't matter which number is first.

### What are the Elements

What are the Elements:

Each element in the matrix can be labeled. We use the word element here, because we can have numbers, fruits, or anything inside. The letter I represents the row, and the letter J represents the column. We always put the row first and the column second just like how we label them. For instance, a1,1 is the element in the first row and the first column, which is 1 in the example above. And, a2,1 is the element in the second row and the first column, which is 4 in the example above.

### Can We Add or Subtract Matrices - Not Always

When Can We Add or Subtract Matrices:

In the example on the left, it might look that we can add/subtract the two matrices, but 2 x 2 and the 2 x 3 matrices do not match. The number of rows are the same, but the number of columns do not. The element b1,3 in matrix B does not have a corresponding element a1,3 in matrix A. Therefore we cannot add these two matrices together.

In the example on the right, we can add/subtract the two matrices since each element has a corresponding element. And our result will be the same type of matrix. In the case above our result will be a 2 x 2 matrix.

### Can We Multiply Matrices - Not Always

Can We Multiply Matrices:

In the example on the left we have two matrices A, which is a 3 x 2 matrix, and B, which is a 2 x 3 matrix. We can multiply this matrices together since the number of columns in matrix A matches the number of rows in matrix B. This means we will have the same number of elements multiplying together. In the case on the left, each square in our resultant matrix will be the result of multiplying the different corresponding elements and adding the results together. For instance in the case on the left, we would multiply a1,1 and b1,2 and a1,2 and b2,2 and add the results to find the answer.

The label of resultant matrix will be the number of rows of the first matrix and the number of columns in the second matrix. So we have 3 x 3 matrix.

In the example on the right the number of columns in matrix A matches the number of rows in matrix B. Even though the numbers flipped from the other matrix, we can still multiply the matrices together, but that isn't always the case. Matrix A is a 2 x 3 and matrix B is a 3 x 2 matrix. The resultant matrix in the case on the right is a 2 x 2 matrix.

Can We Multiply Matrices:

In the example on the left we have two matrices A, which is a 3 x 1 matrix, and B, which is a 2 x 3 matrix. We cannot multiply this matrices together since the number of columns in matrix A does not match the number of rows in matrix B. This means we will not have the same number of elements multiplying together. In the case on the left there is only 1 element but when we try to multiply there are 2 elements. We cannot multiply these two matrices together since we don't have corresponding values. Because in this case, b2,2 does not have a element to match with.

In the example on the right we have flipped A and B and now we multiplying B x A. Now we have 2 x 3 multiplying 3 x 1. This time the number of columns in the first matrix matches the number of rows in the second matrix so we can multiply the matrices together.

The resultant matrix will be a 2 x 1 matrix.

### Diagram of When We Can Multiply

Diagram of When We Can Multiply:

When we multiply two matrices together the number of columns of the first matrix must equal the number of rows in the second matrix.

## Live Worksheet

Here is the link if you prefer.