Question Video: Multiplying a Matrix by a Scalar Mathematics

Video Transcript

If 𝐴 is equal to the one-by-three matrix eight, negative three, one, what is the value of zero 𝐴.

We’re given a matrix 𝐴 and we’re asked to evaluate zero multiplied by 𝐴. Of course, zero is a number, so this is scalar multiplication of our matrix. The first thing we’re going to need to recall is how we multiply a matrix by a scalar. We recall scalar multiplication of a matrix means we multiply every single entry by our scalar. In this case, we’re going to need to multiply every entry by zero. Doing this, we get the one-by-three matrix with entry in row one, column one zero times eight; row one, column two zero times negative three; and row one, column three zero times one. And of course we can evaluate the expressions of all of our entries. We know that zero times eight is equal to zero, zero times negative three is equal to zero, and zero times one is equal to zero. So this gives us the one-by-three matrix with every entry equal to zero, which is our final answer.

Therefore, we were able to show if 𝐴 is equal to the one-by-three matrix eight, negative three, one, then zero 𝐴 will be equal to the one-by-three zero matrix.

We can notice a very useful result from this question. We know that any number multiplied by zero is equal to zero. So, in fact, it didn’t matter what our matrix was. It always would’ve given a matrix with every entry equal to zero. So let’s confirm this result. If we have a matrix of order 𝑚 by 𝑛, and we’ll call this matrix 𝐴, and we’ll call the entry in row 𝑖 and column 𝑗 of matrix 𝐴 𝑎 𝑖𝑗, then if we multiply our matrix by the scalar zero, every entry inside of our matrix should be zero. In other words, this should be equal to the 𝑚-by-𝑛 zero matrix, represented by zero sub 𝑚𝑛.

And in fact, we can prove this result. We could do this by writing the matrix 𝐴 out in matrix notation. However, we already know the entry in row 𝑖 and column 𝑗. So when we multiply our matrix 𝐴 by the scalar zero, we multiply every single entry by zero. In other words, the entry in row 𝑖 and column 𝑗 of this matrix is zero times 𝑎 𝑖𝑗. However, we know for any number, zero times that number will always be equal to zero. In other words, for all 𝑖 and all 𝑗, zero times 𝑎 𝑖𝑗 is equal to zero. So every entry inside of our matrix is equal to zero. In other words, this is equal to the zero matrix of order 𝑚 by 𝑛.

And it’s important to remember we need to keep the order of our matrix because scalar multiplication does not change the order of our matrix. But this is not the only useful result we can get from this definition of scalar multiplication. Another question we can ask is, what is one multiplied by a matrix 𝐴? Remember, when we multiply a matrix by a scalar, we multiply every single entry in that matrix by the scalar. So, in this case, we’re multiplying every entry inside of our matrix by one. Of course, this isn’t going to change the value of any of our entries. So this should be equal to 𝐴.

And in fact, we can prove this using a very similar method to what we did above. When we multiply our matrix 𝐴 by the scalar one, we’re multiplying every single entry in matrix 𝐴 by one. In other words, the entry in row 𝑖 and column 𝑗 is going to be equal to one times 𝑎 𝑖𝑗 because we know the entry in row 𝑖 and column 𝑗 of matrix 𝐴 is 𝑎 𝑖𝑗. And of course, one multiplied by any number is equal to that same number. So one times 𝑎 𝑖𝑗 is going to be equal to 𝑎 𝑖𝑗 for any values of 𝑖 and 𝑗. Therefore, the entries inside of our matrix don’t change. The entry in row 𝑖 and column 𝑗 is just 𝑎 𝑖𝑗. Therefore, we’ve shown for any matrix 𝐴, one 𝐴 is equal to 𝐴.

And that’s another useful result we’re going to need going forward. We want to ask the question, what is negative one 𝐴 equal to? Of course we know how to do this. We multiply every single entry inside the matrix 𝐴 by negative one. So this seems to suggest that negative one times 𝐴 is going to be equal to negative 𝐴. And to fully express this, we’re going to need to recall exactly what we mean by negative 𝐴. The easiest way to do this is to think what happens if you subtract a matrix from itself.

Remember, when you subtract matrices, you do it component-wise. So when we calculate the matrix 𝐴 minus itself, every entry is going to be subtracted from itself. Every entry is going to be equal to zero. And of course, it keeps the dimension 𝑚𝑛. So perhaps a better way to write this equation would be to add our matrix 𝐴 to both sides of the equation. This would give us the equivalent statement 𝐴 plus negative one times 𝐴 is equal to 𝐴 minus 𝐴. And we know exactly what 𝐴 minus 𝐴 is equal to. It’s the zero matrix of order 𝑚 by 𝑛. And we could prove this in exactly the same way we did above.

First, to evaluate negative one multiplied by 𝐴, we need to use scalar multiplication. We multiply every entry inside of our matrix by negative one. So in row 𝑖, column 𝑗, we’ll have 𝑎 𝑖𝑗 plus negative one times 𝑎 𝑖𝑗. And of course we can then simplify this. Negative one multiplied by 𝑎 𝑖𝑗 is going to be equal to negative 𝑎 𝑖𝑗 for any values of 𝑖 and 𝑗. But remember, when we add two matrices together, we do this component-wise. So in row 𝑖, column 𝑗, we’re going to get 𝑎 𝑖𝑗 minus 𝑎 𝑖𝑗, and the number minus itself is equal to zero. So every entry inside of our matrix is going to be zero. This is going to be equal to the 𝑚-by-𝑛 zero matrix.

And there’s one final result which we can get from this definition of scalar multiplication. Before we do this, let’s clear a little bit of space. This result is going to be very similar to our first result. However, this time, instead of multiplying a matrix by the scalar zero, we’re instead going to multiply a zero matrix by any scalar. If we let 𝑘 be any number, then we can consider what happens when we multiply the 𝑚-by-𝑛 zero matrix by 𝑘. Of course, multiplying by the scalar 𝑘 means we multiply every single entry inside of the zero matrix by 𝑘.

But all of the entries inside of the zero matrix are zero. So we’re just going to get 𝑘 multiplied by zero for all of our entries. This is just going to be equal to the 𝑚-by-𝑛 zero matrix. And we could prove this by using a very similar method to what we did before. When we multiply a matrix by a scalar, we need to multiply every single entry inside of our matrix by the scalar. So in this case, we’re going to get in row 𝑖, column 𝑗 𝑘 multiplied by the entry in row 𝑖 and column 𝑗 of the zero matrix. However, every single entry in the zero matrix is zero. So in row 𝑖, column 𝑗, we get 𝑘 times zero which is equal to zero. So this is just equal to the 𝑚-by-𝑛 zero matrix.