This is a matrix question.
R is the matrix (3r 1)
(s 2s)
1. State, in terms of s and r the determinant of R
2. If r=1/3 and s=4 determine the inverse of R
3.State the pair of r and s not including 0, which would make the matrix R a singular matrix
To answer these matrix questions, we need to understand some basic concepts.
1. Determinant of a Matrix:
The determinant of a 2x2 matrix, denoted by |R| or det(R), is calculated using the formula:
det(R) = (3r)(2s) - (s)(1) = 6rs - s = s(6r - 1).
2. Inverse of a Matrix:
To find the inverse of a 2x2 matrix R, we need to use the following formula:
R^(-1) = (1/det(R)) * Adj(R), where Adj(R) represents the adjugate of matrix R.
Now, let's answer the individual questions:
1. The determinant of matrix R, denoted det(R), can be calculated as s(6r - 1).
So, the determinant of R is s(6r - 1).
2. To find the inverse of matrix R, we need to substitute r = 1/3 and s = 4 into the inverse formula.
First, let's calculate the determinant of R using the values given:
det(R) = 4(6 * 1/3 - 1) = 4(2 - 1) = 4 * 1 = 4.
Now, let's calculate the adjugate of matrix R:
Adj(R) = [(2s) (-1r)]
[(-s) (3r)]
Substitute the values r = 1/3, s = 4 and determinant det(R) = 4 into the inverse formula:
R^(-1) = (1/4) * [(2 * 4) (-1 * 1/3)]
[(-4) (3 * 1/3)]
Simplifying, we get:
R^(-1) = (1/4) * [(8) (-1/3)]
[(-4) (1)]
Therefore, the inverse of matrix R is:
R^(-1) = [(2) (-1/12)]
[(-1) (1/4)]
3. A matrix R is said to be singular if its determinant det(R) is equal to 0. According to our previous calculation, the determinant of R is s(6r - 1).
For R to be singular, det(R) must be equal to 0, so we solve the equation:
s(6r - 1) = 0
This equation holds when:
s = 0 (Any value of r)
Therefore, for the matrix R to be singular, the pair of values r and s must be such that s = 0 (and r could be any real number including 0).
I hope this explanation helps you understand how to tackle these matrix questions.