Video: Determining Whether Two Matrices Are Equal

Given that 𝐴 = [3, 3, 3 and 3, 3, 3] and that 𝐡 = [3, 3 and 3, 3], is it true that 𝐴 = 𝐡?

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Video Transcript

Given that 𝐴 is the matrix with first row three, three, three and second row three, three, three and 𝐡 is the matrix with first row three, three and second row three, three, is it true that the matrix 𝐴 is equal to the matrix 𝐡?

Let’s start by recalling what we mean when we say that two matrices are equal. If we have the entry in row 𝑖 and column 𝑗 of matrix 𝐴 is π‘Ž 𝑖𝑗 and the entry in row 𝑖 and column 𝑗 of matrix 𝐡 is 𝑏 𝑖𝑗, then if π‘Ž 𝑖𝑗 is equal to 𝑏 𝑖𝑗 for all of our values of 𝑖 and 𝑗, we say that the matrix 𝐴 is equal to the matrix 𝐡. Otherwise, we say that these matrices are not equal. So to check the two matrices are equal, we need to check that all of their entries are identical.

Let’s start with matrix 𝐴. We can see this has two rows and three columns. So in our case, what would our values of π‘Ž 𝑖𝑗 be? First, our matrix 𝐴 has two rows and three columns, so our values of 𝑖 range from one to two and our values of 𝑗 range from one to three. We can then do something similar for our matrix 𝐡. If 𝑏 π‘˜π‘™ is the entry in matrix 𝐡 in row π‘˜ and column 𝑙, then because our matrix 𝐡 only has two rows and two columns, our values of π‘˜ range from one to two and our values of 𝑙 also range from one to two. But now we can start to see our problem from our definition. The entries must be equal for all possible rows and columns. Our matrix 𝐴 has three columns. However, our matrix 𝐡 only has two columns. So these matrices can’t possibly be equal.

For example, if we highlight the two entries in matrix 𝐴 in column three, by our definition of equality, we would have to have a third column in matrix 𝐡 which is equal to this column. So in this case, the matrix 𝐴 is not equal to the matrix 𝐡.

In fact, we can use exactly the same line of reasoning as we did in this question to deduce that if two matrices have different orders, they can’t be equal. In other words, if they have a different number of rows or columns, they can’t be equal. This means what we’ve shown is for two matrices to be equal, they must have the same number of rows or columns. In other words, they must have the same order.

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