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Question Video: Finding an Unknown Matrix by Applying Operations on Matrices Involving the Zero Matrix Mathematics • 10th Grade

Given that 𝑋 + [−6, −8 and 6, 5] = 0, where 0 is the 2 × 2 zero matrix, find the value of 𝑋.

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Video Transcript

Given that 𝑋 plus the matrix whose elements are negative six, negative eight, six, five equals zero, where zero is the two-by-two zero matrix, find the value of 𝑋.

We begin, of course, by recalling what we mean when we say that a matrix is a zero matrix. This is a square matrix, in other words, a matrix with an equal number of rows and columns whose entries or elements are all equal to zero. And so in this case, the two-by-two zero matrix is the matrix shown. We can therefore rewrite our matrix equation as 𝑋 plus negative six, negative eight, six, five equals zero, zero, zero, zero. Now we’re trying to find the value of 𝑋. So given the equation we have now written, what can we infer about 𝑋? Well, one thing that we know is that if we add a pair of matrices, we simply add the elements. And we can only do that if the matrices are of the same order. We couldn’t add, for example, a two-by-two matrix to a two-by-three matrix, nor could we add a two-by-two matrix to just a number with a single value.

And so, for this matrix equation to make sense, 𝑋 must also be a two-by-two matrix. Now, to solve this matrix equation, we’re actually going to perform a similar set of steps to solving a normal equation. We’re going to subtract this matrix negative six, negative eight, six, five from both sides of the equation. When we subtract it from the left-hand side, of course, we’re just going to end up with the matrix 𝑋. And so 𝑋 is zero, zero, zero, zero minus the matrix whose elements are negative six, negative eight, six, five. Now, just like when we add a pair of matrices and we add their elements to subtract a pair of matrices, we subtract their individual elements.

The element in the first row and first column then will be zero minus negative six. Now, of course, subtracting a negative is the same as adding a positive. So that’s the same as doing zero plus six, which is six. Then, we do zero minus negative eight. And once again, that’s the same as doing zero plus eight, which is eight. We now move on to the elements in the second row, so we work out zero minus six. And, of course, that’s simply negative six. And, finally we’re going to work out zero minus five, and that’s negative five. And so the two-by-two matrix 𝑋 is six, eight, negative six, negative five.

Now, this example illustrates something really important. We see that matrix addition satisfies the additive inverse property. Now, the additive inverse of a number is what you add to a number to create zero, and usually that’s just found by changing the sign of the original number. If we look at our example here, we can see we’ve changed the sign of the individual elements. And so the matrix six, eight, negative six, negative five is the additive inverse of the matrix negative six, negative eight, six, five.

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