Matrices and vectors
A matrix times a vector, the matrix of a rotation or a stretch, matrix products, the determinant and the inverse — linear algebra with numbers.
How to solve it
A matrix is a map of the plane written as a table: its columns are the images of the basis vectors $\mathbf e_1$ and $\mathbf e_2$. Multiplying a matrix by a vector means taking a combination of the columns with the vector's coordinates as coefficients. A product of matrices is two maps in a row.
Step by step
- $A\mathbf v$: multiply the first column by the first coordinate, the second by the second, and add. Or use “row times column”.
- The matrix of a map: find where $\mathbf e_1 = (1,\ 0)$ and $\mathbf e_2 = (0,\ 1)$ go and write the images as columns.
- The product $AB$: the entry in row $i$ and column $j$ is row $i$ of $A$ dotted with column $j$ of $B$.
- A $2 \times 2$ determinant: $ad - bc$. The inverse exists only if it is not zero.
Common mistakes
- Writing the images of the basis vectors as rows instead of columns.
- Assuming $AB = BA$. Matrix multiplication does not commute: the order of the maps matters.
- Changing the signs on the diagonal of the inverse instead of the off-diagonal entries.
- Forgetting to divide by the determinant.