Matrices & Systems — Print
Strip the letters from a system of equations and what is left is a grid of numbers that can be solved mechanically.
What the practice questions cover
These questions use 9 ideas from the course. Follow any of them for a definition, where it is taught, and what it leads to.
- Mean — Two kinds of centre: the balance point, and the middle value.
- Coefficient — The number multiplying a variable — the 3 in 3n.
- System — Two or more equations that must hold at the same time.
- Parallel — Same slope, different intercept.
- Diagonal — A segment joining two non-adjacent vertices.
- Matrix — A rectangular array of numbers, described by its rows then columns.
- Dimensions — Rows × columns. A 2 × 3 matrix has two rows and three columns.
- Identity matrix — The square matrix with ones down the diagonal; it leaves everything unchanged.
- Determinant — A single number computed from a square matrix; zero means it has no inverse.
Write out three equations in three unknowns and look at them.
Every one has an x term, then a y term, then a z term, then an equals sign. Always in that order.
The letters never move. They are column headings, doing no arithmetic at all.
So cross them out. What is left is a rectangle of numbers, and nothing was lost.
That rectangle is a matrix. Once the bookkeeping is separated from the arithmetic, a computer can solve a system of a thousand equations without ever seeing a letter.
Reading and adding
Dimensions are always rows first, then columns. A 2 × 3 matrix has two rows of three.
Adding is done entry by entry, so the two matrices must be exactly the same shape.
Multiplying by a number multiplies every entry, exactly as it does with a vector.
Rows before columns
The entry a23 sits in row 2, column 3. The order is fixed, and every convention in the unit follows it.
Multiplying, and why the shapes must fit
To multiply, take a row of the first and a column of the second, multiply term by term and add — a dot product.
So the row and column must be the same length: the columns of the first must match the rows of the second.
A 2 × 3 times a 3 × 4 works and gives a 2 × 4. The inside numbers agree, and the outside ones give the answer.
Order changes the answer
AB and BA are usually different, and often one of them does not even exist. Matrix multiplication is not commutative, unlike every product before this unit.
The determinant
For a 2 × 2 matrix, det = ad - bc. It measures how much the matrix stretches area.
A determinant of zero means the matrix collapses the plane onto a line or a point, and flattening cannot be undone.
So a zero determinant is exactly the signal that no inverse exists.
What zero means for a system
A system whose determinant is zero has either no solution or infinitely many — never exactly one. The geometry and the algebra agree.
Inverses, and solving a system in one step
If Ax = b, then x = A-1b — the whole system solved by one multiplication.
For 2 × 2, A-1 = 1ad-bc[d-b-ca]: swap the diagonal, flip the signs off it, divide by the determinant.
This is why a calculator can solve a system instantly, and why it refuses when the determinant is zero.
Check by multiplying back
Multiply A by your A-1 and you should get the identity. Anything else means a sign slipped, and the check costs one line.
Row reduction
Write the coefficients and the constants as one augmented matrix, then use row operations to simplify.
You may swap rows, multiply a row by a non-zero number, or add a multiple of one row to another — each is a legal move on the equations themselves.
Drive it towards ones on the diagonal and zeros below, and the last row reads off a value you can back-substitute.
Nothing new is happening
Every row operation is something you would do to the equations anyway. The matrix just saves rewriting the letters each time.
Worked examples
Multiplying
Find [1230][45].
- Shapes: 2 × 2 times 2 × 1. The inside 2s agree, so the answer is 2 × 1. Check before calculating.
- Row one with the column: 1(4)+2(5) = 14. Multiply matching entries and add.
- Row two with the column: 3(4)+0(5) = 12. Same move, next row.
[1412]
A determinant
Find the determinant of [5324] and say whether it has an inverse.
- det = ad - bc = 5(4) - 3(2). Down the diagonal minus up the diagonal.
- = 20 - 6 = 14. Not zero.
- A non-zero determinant means the inverse exists. The matrix does not flatten the plane.
14; yes
Solving a system
Use the determinant to decide how many solutions 2x+3y=7, 4x+6y=1 has.
- The coefficient matrix is [2346]. Strip the letters.
- det = 2(6)-3(4) = 12-12 = 0. Zero, so there is no unique solution.
- The second equation is twice the first on the left but not on the right, so the lines are parallel. Parallel and distinct means no solution at all.
No solution
Words to know
- Matrix
- A rectangular array of numbers, described by its rows then columns.
- Dimensions
- Rows × columns. A 2 × 3 matrix has two rows and three columns.
- Identity matrix
- The square matrix with ones down the diagonal; it leaves everything unchanged.
- Determinant
- A single number computed from a square matrix; zero means it has no inverse.
- Augmented matrix
- The coefficients with the constants attached as a final column.
Mistakes to watch for
Assuming AB = BA. Matrix multiplication depends on order, and often only one of the two products is even defined. Checking the shapes first catches this immediately.
Multiplying entry by entry. That is how addition works, not multiplication — each entry of a product is a whole row dotted with a whole column.
Mismatching dimensions. A 2 × 3 can multiply a 3 × 4 but not a 4 × 3. The inside numbers must agree, and no arithmetic can rescue it if they do not.
Dividing by a zero determinant. A determinant of zero means no inverse exists, so the system has either no solution or infinitely many. Pressing on gives a division by zero.
Forgetting the sign flips in the 2 × 2 inverse. The diagonal entries swap and the other two change sign. Multiplying back to check the identity catches any slip in one line.
The word "matrix" in this sense is Sylvester’s, from 1850 — he meant it as a womb from which determinants are born.
Google’s original PageRank was one enormous matrix problem: find the vector that the web’s link matrix leaves unchanged.
Rotating, scaling and skewing an image are all matrix multiplications, which is why a graphics chip is essentially a machine for multiplying small matrices very fast.
Problem 1
Give the dimensions of a matrix with 3 rows and 5 columns, in the form rows x columns.
Worked answer
3x5
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Problem 2
Add [1234] and [5012]. Give the top-left entry.
Worked answer
6
1+5 = 6
Problem 3
Multiply [2314] by 5. Give the bottom-right entry.
Worked answer
20
4 × 5 = 20
Problem 4
Find the determinant of [3124].
Worked answer
10
3(4)-1(2) = 12-2 = 10
Problem 5
Find the determinant of [2613].
Worked answer
0
2(3)-6(1) = 6-6 = 0
Problem 6
Can a 2 × 3 matrix multiply a 2 × 4 matrix?
- No — the inside numbers disagree
- Yes, giving 2 × 4
- Yes, giving 3 × 2
- Only if both are square
Worked answer
No — the inside numbers disagree
3 ≠ 2, so the rows and columns cannot pair up
Problem 7
A 2 × 3 matrix multiplies a 3 × 4 matrix. Give the dimensions of the result, as rows x columns.
Worked answer
2x4
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Problem 8
Find the top entry of [1230][45].
Worked answer
14
1(4)+2(5) = 14
Problem 9
Find the bottom entry of the same product.
Worked answer
12
3(4)+0(5) = 12
Problem 10
Is AB always equal to BA for matrices?
- No
- Yes
- Only for 2 × 2
- Only when both are invertible
Worked answer
No
Matrix multiplication depends on order
Often only one of the two products even exists
Problem 11
A 2 × 2 matrix has determinant 0. How many inverses does it have?
Worked answer
0
Zero determinant means no inverse exists
Problem 12
For [4726], find the determinant.
Worked answer
10
4(6)-7(2) = 24-14 = 10
Problem 13
For the system 2x+3y=7, 4x+6y=1, find the determinant of the coefficient matrix.
Worked answer
0
2(6)-3(4) = 0
So there is no unique solution
Problem 14
A system has coefficient determinant 0 and the equations are not multiples of each other. How many solutions?
- None
- Exactly one
- Infinitely many
- Two
Worked answer
None
The lines are parallel and distinct
Parallel lines never meet
Problem 15
What is the top-left entry of the 2 × 2 identity matrix?
Worked answer
1
[1001]
Problem 16
Find k so that [k436] has no inverse.
Worked answer
2
6k-12 = 0
k = 2