Video Transcript
If negative three 𝑋 plus the
matrix negative three, zero, nine, 12 is equal to the zero matrix, then 𝑋 equals
what. Is it (A) one, zero, negative
three, negative four; (B) negative one, zero, three, four; (C) three, zero, negative
nine, negative 12; or (D) negative three, zero, nine, 12?
We recall that the zero matrix has
all elements equal to zero. Therefore, the two-by-two zero
matrix is equal to zero, zero, zero, zero. There are lots of ways of solving
this problem. We could divide through by three or
negative three. However, in this example, we will
let the matrix 𝑋 be the two-by-two matrix with elements 𝑎, 𝑏, 𝑐, 𝑑. In order to multiply any matrix by
a scalar, we simply multiply each of the elements or components by that scalar. This means that the matrix negative
three 𝑋 has elements negative three 𝑎, negative three 𝑏, negative three 𝑐,
negative three 𝑑. Adding the matrix negative three,
zero, nine, 12 to this will give us the zero matrix.
We can then set up four linear
equations by comparing the corresponding components or elements. Firstly, negative three 𝑎 plus
negative three is equal to zero. We can add three to both sides such
that negative three 𝑎 is equal to three. Dividing through by negative three
gives us 𝑎 is equal to negative one. Next, we can look at the top-right
elements. This gives us negative three 𝑏
plus zero is equal to zero. As negative three 𝑏 is equal to
zero, dividing both sides by negative three gives us 𝑏 is equal to zero. We can repeat this process for the
bottom row, giving us values of 𝑐 and 𝑑 equal to three and four. The matrix 𝑋 is therefore equal to
negative one, zero, three, four. We can therefore conclude that the
correct answer is option (B).
