Question Video: Solving a Simple Equation with Determinants Mathematics • First Year of Secondary School

Video Transcript

If the determinant of the matrix one, 𝑥, 𝑥, three is equal to the determinant of the matrix two, one, four, three, then 𝑥 equals blank.

We recall that the determinant of any two-by-two matrix of the form 𝑎, 𝑏, 𝑐, 𝑑 is equal to 𝑎𝑑 minus 𝑏𝑐. Let’s begin by looking at the matrix one, 𝑥, 𝑥, three. Multiplying the top-left and bottom-right values gives us three. Multiplying the top-right and bottom-left values gives us 𝑥 squared. This means the determinant of this matrix is three minus 𝑥 squared. We can repeat this process for our second matrix. Two multiplied by three is equal to six. One multiplied by four is equal to four. As six minus four is equal to two, the determinant of this matrix is two.

In order to calculate the value of 𝑥, we need to solve the equation three minus 𝑥 squared is equal to two. We can add 𝑥 squared and subtract two from both sides of this equation. This gives us three minus two is equal to 𝑥 squared. As 𝑥 squared is equal to one, we can square root both sides of this equation. The square root of one is equal to one. This means that 𝑥 could be positive one or negative one, as after square rooting a number, our answer can be positive or negative. If the determinant of the two matrices are equal, then 𝑥 can be equal to one or negative one.

We could check this answer by substituting these values back into the expression three minus 𝑥 squared. Squaring one or negative one gives us an answer of one. Therefore, we have three minus one. As this is equal to two, which was the determinant of the second matrix, we know that our answers are correct. 𝑥 can be equal to one or negative one.