For 2x² + 3x + 5, how many real roots are there?

None. D = b² − 4ac = 9 − 40 = −31. Negative discriminant means the square root has no real value, so the parabola never reaches the x-axis. Since a > 0 it opens upward and sits entirely above it.

A 2×2 matrix has determinant 0. What does that mean geometrically?

It squashes the plane onto a line or point. Determinant is the area scale factor. Scaling area by 0 means the output has no area at all — the whole 2D plane has been collapsed into a 1D line. That loses information permanently, which is exactly why singular matrices cannot be inverted.

The vertex of y = ax² + bx + c sits at x = ?

−b / 2a. The parabola is symmetric, so its turning point sits exactly midway between the two roots. Averaging the quadratic-formula roots, the ± √D parts cancel and you are left with −b/2a. It works even when there are no real roots.

Matrix M sends î to (0, 1) and ĵ to (−1, 0). What does M do?

Rotates the plane 90° anticlockwise. The right-pointing arrow now points up, and the up-pointing arrow now points left. Every vector has swung a quarter turn anticlockwise. Its determinant is (0)(0) − (−1)(1) = 1, confirming area is preserved, as a rotation must.

Why is AB generally not equal to BA for matrices?

Because they represent transformations, and the order you apply transformations changes the outcome. Matrix products are compositions of transformations. Rotating then shearing genuinely lands somewhere different from shearing then rotating. The non-commutativity is a real geometric fact, not an algebraic accident.

If det(A) = 5 and you apply A to a shape of area 3, the new area is:

15. The determinant multiplies area, and it does so for every shape, not just the unit square. 3 × 5 = 15. If the determinant had been −5 the area would still be 15, but the shape would come out mirror-imaged.

Algebra & Matrices — Mathematics Notes, Worked Examples and Quiz

Quadratics: one shape, three questions

Every quadratic ax² + bx + c draws the same shape — a parabola. Only three things ever change: which way it opens (sign of a), where its turning point sits, and whether it crosses the x-axis.

That last question is answered entirely by one number, the discriminant.

x = ( −b ± √(b² − 4ac) ) / 2aD = b² − 4acD > 0 → two roots · D = 0 → one root · D < 0 → none

Drag c upward until the curve lifts clear of the x-axis. Watch D go negative at the exact moment the roots vanish.

Why the discriminant works

Look inside the square root: √(b² − 4ac). If that quantity is negative you are asking for the square root of a negative number, which has no real answer — so there is no real x where the curve touches zero.

If it is exactly zero the ± does nothing, both roots collapse into one, and the parabola kisses the axis at its vertex. If it is positive you get two genuinely different answers.

A matrix is a machine that moves space

Stop thinking of a matrix as a box of numbers. A 2×2 matrix is an instruction for rearranging the whole plane — stretch it, squash it, rotate it, shear it.

The instruction is written down in the simplest possible way: the columns tell you where the two basis arrows land. Column one is the new position of î (originally pointing one unit right). Column two is the new position of ĵ (originally one unit up). Everything else follows.

Set b to 0 and c to 0 — pure stretching. Now set a=0, b=−1, c=1, d=0 — a clean 90° rotation. The amber square is the unit square, and its area is the determinant.

The determinant is an area

The unit square has area 1. After the transformation it becomes a parallelogram. The determinant is simply how much the area got multiplied by.

A determinant of 3 means everything triples in area. A determinant of 0 means the plane has been flattened onto a line — information is destroyed, and the matrix has no inverse. A negative determinant means space got flipped over, like turning a page.

det [ a b ; c d ] = ad − bczero determinant → singular → not invertible

Order matters

Matrix multiplication is not commutative: AB ≠ BA in general. This is not a quirk of the notation — it is physically true. Rotate a book 90° then flip it, versus flip it then rotate 90°, and it ends up facing differently. Try it with an actual book.

Matrices: order, types, and what can be added to what

A matrix is a rectangular array of numbers. Its order is written rows × columns, so a matrix with 2 rows and 3 columns is of order 2 × 3, and an entry is named a_ij for the element in row i and column j.

Two matrices can be added only if they have the same order, and you simply add corresponding entries. Multiplication is different and stricter: A × B exists only when the number of columns of A equals the number of rows of B, and the result has the rows of A and the columns of B.

Type	Meaning
Row matrix	one row only, order 1 × n
Column matrix	one column only, order m × 1
Square matrix	equal numbers of rows and columns
Diagonal matrix	square, with every off-diagonal entry zero
Identity matrix I	diagonal, with 1 on the diagonal; AI = IA = A
Null matrix O	every entry zero
Transpose Aᵗ	rows and columns interchanged

Matrix multiplication is not commutative

AB and BA are usually different, and one of them may not even exist. Never cancel matrices the way you cancel numbers, and never assume you may swap the order of a product — a large part of the marks in this chapter is for keeping the order as given.

Solving simultaneous equations with a matrix

Two linear equations can be written as a single matrix equation AX = B, where A holds the coefficients, X the unknowns and B the constants. There are then two standard routes to the solution, and the paper accepts either.

The inverse matrix method multiplies both sides by A⁻¹, giving X = A⁻¹B. Cramer's rule avoids the inverse: replace a column of A by B and take a ratio of determinants.

A⁻¹ = (1 / |A|) × adj Awhere |A| = ad − bcX = A⁻¹ Bx = |Aₓ| / |A|y = |A_y| / |A|(Cramer's rule)both methods fail when |A| = 0 — and that failure is itself the answer to "why has this system no unique solution"

Worked example

Solve 2x + 3y = 13 and x − y = 1 using Cramer's rule.

Coefficient determinant |A| = (2)(−1) − (3)(1) = −5.Non-zero, so a unique solution exists.
Replace the x column by the constants: |Aₓ| = (13)(−1) − (3)(1) = −16.
x = |Aₓ| / |A| = −16 / −5 = 3.2.Both determinants are negative, so the ratio is positive. A sign slip in either one shows up immediately as an answer that fails the check at the end.
Replace the y column: |A_y| = (2)(1) − (13)(1) = −11, so y = −11/−5 = 2.2.Both values check in both original equations, which is the verification the mark scheme wants.

x = 3.2, y = 2.2

What a zero determinant is telling you

If |A| = 0 the matrix is singular and has no inverse, and Cramer's rule divides by zero. Geometrically the two lines are parallel or identical, so the system has either no solution or infinitely many — never exactly one. Stating that is the answer to the question, not a sign that you have gone wrong.

Algebra & Matrices