Matrices and Transformations
What the domain tests
Matrices and transformations tests matrix arithmetic, determinants, inverses, linear systems and geometric transformations. MSCE Additional Mathematics expects you to connect a matrix with a transformation and to preserve the order of multiplication, since matrix products are generally not commutative.
Core rules
- Matrices add only when they have the same order. For AB, the number of columns of A must equal the number of rows of B; generally AB≠BA.
- For a 2×2 matrix A=(a b; c d), detA=ad-bc. A^{-1}=1/(ad-bc)(d -b; -c a), provided detA≠0.
- To solve AX=B, multiply on the left by A^{-1}: X=A^{-1}B.
- Transformation matrices act on column vectors. Reflection in the x-axis: (1 0;0 -1); reflection in the y-axis: (-1 0;0 1); rotation 90° anticlockwise about O: (0 -1;1 0); enlargement scale k: (k 0;0 k).
- The determinant gives the area scale factor; a negative determinant indicates orientation reversal.
Worked example
Solve 2x+y=7 and x+3y=11 using matrices. A=(2 1;1 3), so detA=6-1=5. A^{-1}=1/5(3 -1;-1 2). Then X=A^{-1}(7;11)=1/5(21-11,-7+22)=(2,3). Thus x=2 and y=3.
Traps
Multiplying matrices in the wrong order; forgetting to divide by the determinant; using ad+bc for the determinant; applying a transformation matrix on the right of a column vector.