Video: Applying Operations on Matrices Involving Scalar Multiplication to Find an Unknown Matrix

Given that π‘₯ = [ βˆ’3, βˆ’2 and 1, 5 and βˆ’8, βˆ’8 ], 𝑦 = [ βˆ’1, 8 and βˆ’9, βˆ’9 and 7, βˆ’2 ], 𝑧 = [ 3, βˆ’8 and βˆ’7, 0 and βˆ’8, 5 ], what is the matrix 3π‘₯ + 𝑦 βˆ’ 3𝑧?

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Video Transcript

Given that π‘₯ is equal to negative three, negative two, one, five, negative eight, negative eight. 𝑦 is equal to negative one, eight, negative nine, negative nine, seven, negative two. And 𝑧 is equal to three, negative eight, negative seven, zero, negative eight, five. What is the matrix three π‘₯ plus 𝑦 minus three 𝑧?

Our three matrices π‘₯, 𝑦, and 𝑧 are all of the same order. That is, they have the same number of rows and the same number of columns. And so working out three π‘₯ plus 𝑦 minus three 𝑧 is fairly straightforward. Let’s begin with three π‘₯.

Three π‘₯ is three times negative three, negative two, one, five, negative eight, negative eight. And to multiply a matrix by a constant scalar, we simply multiply each of its elements. Three multiplied by negative three is negative nine. Three multiplied by negative two is negative six. Three times one is three. This element here is three times five, which is 15. And the elements in our bottom row are three times negative eight each. So that’s negative 24. And so we know the value of three π‘₯.

We’re now going to work out the value of three 𝑧. It’s three times each of the elements in the matrix 𝑧, three, negative eight, negative seven, zero, and negative eight, five. The elements in our first row are three times three and three times negative eight. So that’s nine and negative 24. We multiply negative seven and zero by three to get negative 21 and zero. And the final two elements are negative 24, 15. So three π‘₯ plus 𝑦 minus three 𝑧 is now negative nine, negative six, three, 15, negative 24, negative 24 plus negative one, eight, negative nine, negative nine, seven, negative two minus nine, negative 24, negative 21, zero, negative 24, 15.

Now, one of the properties of matrices of the same orders if we’re adding or subtracting those matrices, we simply add or subtract their respective elements. So the first element in our matrix is the sum of negative nine and negative one minus nine. Well, that’s negative 19. We then have negative six plus eight minus negative 24, which is 26. Three plus negative nine minus negative 21 is 15. 15 plus negative nine minus zero is six. Then, for the bottom-left element, we have negative 24 plus seven minus negative 24, which is simply seven. And then, our very final element is negative 24 plus negative two minus 15, which is negative 41.

And so the matrix three π‘₯ plus 𝑦 minus three 𝑧 is negative 19, 26, 15, six, seven, negative 41.

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