Why bother with matrices?
Ever wondered how computer graphics rotate a picture or how engineers solve big systems of equations? Matrices are the backstage heroes that make it happen.
💡 In Simple Words: A matrix is just a neat table of numbers. The determinant is a single special number you can pull out of a square table, telling you things like whether the table can be reversed.
What is a Matrix?
A matrix (plural: matrices) is a rectangular arrangement of numbers called elements. We write the size as "rows × columns". For example, a 2×3 matrix has 2 rows and 3 columns.
Matrix addition and subtraction
To add or subtract matrices, they must be the same size. Just line‑up the corresponding elements and add (or subtract) them.
\[\begin{bmatrix}1 & 2\\3 & 4\end{bmatrix} + \begin{bmatrix}5 & 6\\7 & 8\end{bmatrix}=\begin{bmatrix}1+5 & 2+6\\3+7 & 4+8\end{bmatrix}=\begin{bmatrix}6 & 8\\10 & 12\end{bmatrix}\]Matrix multiplication
Multiplying isn’t element‑by‑element. Instead, you take rows of the first matrix and columns of the second, multiply pairwise, then add up the products. The inner dimensions must match: if A is m×n and B is n×p, the result is m×p.
\[\begin{bmatrix}1 & 2\\3 & 4\end{bmatrix}\times\begin{bmatrix}5 & 6\\7 & 8\end{bmatrix}=\begin{bmatrix}1\cdot5+2\cdot7 & 1\cdot6+2\cdot8\\3\cdot5+4\cdot7 & 3\cdot6+4\cdot8\end{bmatrix}=\begin{bmatrix}19 & 22\\43 & 50\end{bmatrix}\]Determinant: The Magic Number
The determinant only exists for square matrices (same number of rows and columns). Think of it as a single number that captures the “volume‑changing” power of the matrix. If the determinant is zero, the matrix can’t be inverted – it’s like a flat pancake that can’t hold water.
How to find a 2×2 determinant
For a matrix \(\begin{bmatrix}a & b\\c & d\end{bmatrix}\), the determinant is ad − bc. Easy, right?
\[\det\begin{bmatrix}3 & 5\\2 & 7\end{bmatrix}=3\cdot7-5\cdot2=21-10=11\]Determinant of a 3×3 matrix
We use the “rule of Sarrus” or expand by minors. Expanding by minors means picking a row (or column), then for each element, multiply it by its cofactor (the signed determinant of the smaller matrix left after removing that element’s row and column).
\[\det\begin{bmatrix}a & b & c\\d & e & f\\g & h & i\end{bmatrix}=a\cdot\det\begin{bmatrix}e & f\\h & i\end{bmatrix}-b\cdot\det\begin{bmatrix}d & f\\g & i\end{bmatrix}+c\cdot\det\begin{bmatrix}d & e\\g & h\end{bmatrix}\]That looks messy, so let’s see it in action.
\[\det\begin{bmatrix}1 & 2 & 3\\0 & 4 & 5\\1 & 0 & 6\end{bmatrix}=1\cdot(4\cdot6-5\cdot0)-2\cdot(0\cdot6-5\cdot1)+3\cdot(0\cdot0-4\cdot1)\]\n=1\cdot24-2\cdot(-5)+3\cdot(-4)=24+10-12=22\]Quick flowchart for finding a determinant
Key properties of determinants
| Property | What it means |
|---|---|
| Row swap changes sign | If you exchange two rows, the determinant flips from positive to negative (or vice‑versa). |
| Row multiplication | Multiplying a row by a constant k multiplies the whole determinant by k. |
| Row addition | Adding a multiple of one row to another leaves the determinant unchanged. |
| Triangular matrix | For an upper or lower triangular matrix (all zeros below or above the diagonal), the determinant is the product of the diagonal entries. |
Using matrices to solve linear equations
Write the system \(AX = B\) where A is the coefficient matrix, X the column of variables, and B the constants. If \(\det A \neq 0\), the system has a unique solution: \(X = A^{-1}B\) (where \(A^{-1}\) is the inverse matrix). In exams, you’ll often use Cramer's rule – replace one column of A with B, take its determinant, then divide by \(\det A\).
Cramer's rule example
\[\begin{cases}2x + 3y = 5\\4x - y = 1\end{cases}\]\nA=\begin{bmatrix}2 & 3\\4 & -1\end{bmatrix},\;B=\begin{bmatrix}5\\1\end{bmatrix}\]\n\det A = 2\cdot(-1)-3\cdot4 = -2-12 = -14\n\det A_x = \begin{bmatrix}5 & 3\\1 & -1\end{bmatrix}=5\cdot(-1)-3\cdot1 = -5-3 = -8\n\det A_y = \begin{bmatrix}2 & 5\\4 & 1\end{bmatrix}=2\cdot1-5\cdot4 = 2-20 = -18\n x = \frac{-8}{-14}=\frac{4}{7},\; y = \frac{-18}{-14}=\frac{9}{7}\]📝 Likely Exam Questions
- Find the determinant of \(\begin{bmatrix}3 & 0 & 2\\1 & -1 & 4\\0 & 5 & -2\end{bmatrix}\).
Answer: Using expansion by minors, determinant = 3·((-1)(-2)-4·5) - 0·(...) + 2·(1·5-(-1)·0) = 3·(2-20) + 2·5 = 3·(-18)+10 = -54+10 = -44. - State three properties of determinants and give a short example for each.
Answer: (i) Swapping rows changes sign – swapping rows of \([1\ 2;3\ 4]\) changes det from -2 to 2. (ii) Multiplying a row by k multiplies det by k – multiply first row of \([1\ 2;3\ 4]\) by 3, det becomes -6. (iii) Adding a multiple of one row to another leaves det unchanged – R2 → R2 + 2R1 leaves det = -2. - Solve the system using matrices: \(x + 2y = 7,\; 3x - y = 5\).
Answer: A = \([1\ 2;3\ -1]\), B = \([7\ 5]\). \(\det A = 1·(-1)-2·3 = -1-6 = -7\). \(\det A_x = \([7\ 2;5\ -1]\) = 7·(-1)-2·5 = -7-10 = -17\). \(\det A_y = \([1\ 7;3\ 5]\) = 1·5-7·3 = 5-21 = -16\). So \(x = (-17)/(-7)=17/7,\; y = (-16)/(-7)=16/7\). - Explain why a matrix with determinant zero cannot have an inverse.
Answer: The inverse involves dividing by the determinant. If det = 0, division is impossible, meaning the matrix compresses space to a lower dimension and cannot be undone.