Question Video: Using Determinants to Find the Missing Value in a 3 × 3 Matrix Mathematics

What value can 𝑘 not take if the rank of the matrix 𝐴 = [7, 4, −15 and 22, 𝑘, −24 and −9, 15, −21] is 3?


Video Transcript

What value can 𝑘 not take if the rank of the matrix 𝐴 equal to seven, four, negative 15, 22, 𝑘, negative 24, negative nine, 15, negative 21 is three?

In this question, we are told that the rank of a three-by-three matrix is three. We begin by recalling that the rank of an 𝑛-by-𝑛 matrix can only be equal to 𝑛 if the determinant is nonzero. Since we need to find the value that 𝑘 cannot take, this is the value where the determinant of 𝐴 is equal to zero. We can calculate the determinant of any three-by-three matrix by expanding over any row or column in the matrix. And in this question, we will use the most common method of expanding over the top row.

We begin by multiplying the first element in this row, seven, by the determinant of the two-by-two matrix 𝑘, negative 24, 15, negative 21. And this is equal to seven multiplied by negative 21𝑘 plus 360. To calculate the expression in the parentheses, we multiply 𝑘 by negative 21 and then subtract negative 24 multiplied by 15. Next, we multiply negative four by the determinant of the two-by-two matrix 22, negative 24, negative nine, negative 21. This is equal to negative four multiplied by negative 462 minus 216. Finally, we add negative 15 multiplied by the determinant of the two-by-two matrix 22, 𝑘, negative nine, 15. This is the same as subtracting 15 multiplied by 330 plus nine 𝑘.

We now have an expression for the determinant, which we can simplify by distributing the parentheses. The determinant of matrix 𝐴 is equal to negative 147𝑘 plus 2520 plus 2712 minus 4950 minus 135𝑘, which in turn simplifies to negative 282𝑘 plus 282. Setting the determinant equal to zero, we can solve for 𝑘 by adding 282𝑘 to both sides. Dividing through by 282, we have 𝑘 is equal to one.

This means that if 𝑘 is equal to one, the determinant of matrix 𝐴 is zero. And since this would lead to a rank that is not equal to three, the value that 𝑘 cannot take is one.

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