Given that matrix zero, negative two, two 𝑎 plus four, two 𝑏 plus nine equals the matrix zero, negative two, two, negative five, determine the values of 𝑎 and 𝑏.
The key piece of information is going to help us answer this question is recognising that each of these two matrices are equal to one another. And we say that two matrices are equal if they have the same order and their corresponding elements are identical. Now, in this case, we see that the elements in the top row are indeed equal. So we need to find the values of 𝑎 and 𝑏 such that the elements in the bottom row are also equal.
We can, therefore, say that two 𝑎 plus four must be equal to two. And two 𝑏 plus nine must be equal to negative five. And now, we have two simple linear equations that we can solve for 𝑎 and 𝑏, respectively. In our equation for 𝑎, we’re going to solve by subtracting four from both sides. And we find that two 𝑎 is equal to negative two. We then divide through by two. And we see that 𝑎 is equal to negative one. Now, it’s really sensible to check this answer by substituting 𝑎 equals negative one into our original expression that was two 𝑎 plus four. We get two times negative one plus four, which is equal to two, as required.
We’ll now move on to our equation for 𝑏. This time, we’re gonna subtract nine from both sides. Now, be careful. Negative five minus nine means we move further down the number line. So negative five minus nine is negative 14. And we find that two 𝑏 is equal to negative 14. We then divide through by two and we find 𝑏 is equal to negative seven. Once again, we substitute our answer by substituting 𝑏 equals negative seven into this expression. That’s two times negative seven plus nine, which is indeed negative five, as we were hoping.
And so, we see that for these two matrices to be equal, 𝑎 must be equal to negative one and 𝑏 must be equal to negative seven.