Question 1 Foundation
Find .
Answer.
.
Matrices and Systems of Linear Equations
Original questions written at exam standard. Work each one before you open the solution.
Difficulty
Question 2 Foundation
What is for an invertible matrix ?
Answer. The identity matrix
This is the definition of an inverse, and it is the standard check on any inverse you calculate.
Question 3 Standard
Find .
Answer.
. Expanding along the first row, the zero removes a whole term.
Question 4 Standard
Under what condition does a square matrix have an inverse?
Answer. Its determinant is non-zero.
The inverse formula divides by the determinant, so a zero determinant makes the matrix singular.
Question 5 Stretch
The system , has determinant zero. How many solutions does it have?
Answer. None
The left sides are proportional but the constants are not: doubling the first gives , contradicting 7. The equations are inconsistent.
Question 6 Stretch
In row reduction, what does a row of the form tell you?
Answer. The system has no solution.
That row asserts , which is impossible. A row of all zeros including the constant would instead mean infinitely many solutions.