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.
