Video: Calculating Higher Powers of Matrices

Given the matrix 𝐴 = [ 4, 0 and −3, 7 ], calculate 𝐴³ − 3𝐴².

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Video Transcript

Given the matrix 𝐴 equals four, zero, negative three, seven, calculate 𝐴 cubed minus three 𝐴 squared.

Well, the first thing in this question that we’re going to want to calculate is 𝐴 squared. And this is gonna be equal to the matrix four, zero, negative three, seven multiplied by the matrix four, zero, negative three, seven. And we know that we can multiply these two matrices because we have the same number of rows in matrix one as there are a number of columns in matrix two. And this is a condition required when we’re multiplying matrices.

Now, to work out what the elements are going to be in our new matrix, then what we need to do is multiply the corresponding elements from the first row in the first matrix and the first column in the second matrix and then add them together. And when we do this, this will give us our top-left element in our 𝐴 squared matrix. So, what we’re gonna have is four multiplied by four. Because it’s the first element in the first row of the first matrix multiplied by the first element in the first column of the second matrix. Then, add zero multiplied by negative three cause that’s the second element in the first row of our first matrix multiplied by the second element in the first column of our second matrix.

So, now, for the next element in the top row of our 𝐴 squared matrix, what we’re gonna do is we’re gonna multiply the elements in our first row of our first matrix by the elements in our second column of our second matrix. So, first of all, we have four multiplied by zero, and then we add to it zero multiplied by seven. So then, we move on to our bottom row. And we’re gonna have negative three multiplied by four plus seven multiplied by negative three. And then, for the final element, we’re gonna have negative three multiplied by zero plus seven multiplied by seven.

Okay, great, so that’s each of our elements completed. So, now, what we can do is calculate these. So, we start with 16 add zero, which is just 16. Then, we have zero add zero, which is just zero. Then, negative 12 add negative 21. Well, this is the same as negative 12 minus 21, which gives us negative 33. And then, finally, we have zero add 49, which is gonna give us 49. So, that means the matrix for 𝐴 squared is 16, zero, negative 33, and 49. So, now, what’s the next stage?

Well, now, what we want to find out is what is 𝐴 cubed. Well, 𝐴 cubed is gonna be equal to 𝐴 squared multiplied by 𝐴, so the matrix 16, zero, negative 33, 49 multiplied by the matrix four, zero, negative three, seven. And then, to calculate this, what we do is do it in exactly the same way as we did for the previous part where we did 𝐴 squared. So, we’re gonna do is 16 multiplied by four and zero multiplied by negative three. So, that’s our first element.

And then, we’re gonna have 16 multiplied by zero add zero multiplied by seven. Then, moving on to the bottom row, what we’re gonna get is negative 33 multiplied by four plus 49 multiplied by negative three. And then, the final element is gonna be negative 33 multiplied by zero plus 49 multiplied by seven. So, great, now, what we need to do is find out what these are, so calculate each element.

So, the first element is gonna be 64 cause it’s 64 add zero. And then, we have zero add zero, which is just zero. So then, next, we’ve got negative 279. And that’s cause we’ve got negative 132 add negative 147. Well, that’s the same as negative 132 minus 147. And then, finally, we have 343. And this is cause we had a zero add 343. Okay, great, so we’ve now found 𝐴 cubed. Now, there’s just one more part we need to find before we put it all back together.

And that final part that we need to find is three 𝐴 squared. And this is just gonna be equal to three multiplied by our matrix 16, zero, negative 33, 49. And to do that, what we do is we multiply each of our elements by three. And when we do that, we get 48, zero, negative 99, 147. Okay, great, so we now have all the parts we need to calculate 𝐴 cubed minus three 𝐴 squared.

So, now that we have 𝐴 squared, 𝐴 cubed, and three 𝐴 squared, we can see which parts we’ll need for our calculation. Well, we’re only gonna need 𝐴 cubed and three 𝐴 squared. So, what we’re gonna have is the matrix 64, zero, negative 279, 343 minus the matrix 48, zero, negative 99, 147. And what we need to do with this is we just subtract the corresponding elements from each of our matrices.

So, what we’re gonna get is our first element is gonna be 64 minus 48. Next element is zero minus zero. Then, we’re gonna have negative 279 minus negative 99 and then 343 minus 147, which is gonna give us the matrix 16, zero, negative 180, 196. So therefore, we can say that given the matrix 𝐴 equals four, zero, negative three, seven, then 𝐴 cubed minus three 𝐴 squared is equal to the matrix 16, zero, negative 180, 196.

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