Matrices and Determinants: Operations and Linear Systems
What you must command
Matrix questions test careful bookkeeping. Know matrix notation and order, and when addition, subtraction and multiplication are defined. Matrix multiplication requires the columns of the first to match the rows of the second, and the product's order comes from the outer dimensions. Know the identity matrix, the transpose, and that matrix multiplication is not generally commutative. For 2 by 2 matrices, know the determinant ad - bc, the condition for a matrix to be singular, and the inverse formula using the adjugate with the sign pattern. Understand how matrices represent simultaneous linear equations and how the inverse solves them.
Traps that catch candidates
- Check orders before multiplying; incompatible orders mean the product is undefined.
- AB is usually not equal to BA, so keep the given order.
- In the inverse, swap the leading diagonal entries and change the signs of the other two, then divide by the determinant.
- A zero determinant means no inverse exists, so the equations have no unique solution.
- Adding matrices requires identical orders; do not add a 2 by 3 to a 3 by 2.
- Watch arithmetic signs; determinant errors propagate through the whole inverse.
Worked example
For A with rows (3, 1) and (2, 4), the determinant is 3 times 4 minus 1 times 2 = 10. The inverse is one tenth of the matrix with rows (4, -1) and (-2, 3).
Rule of thumb
Before any matrix work, write the order of every matrix in the question. Most errors are dimension or sign slips, not conceptual gaps.