JENECONKMaths Universe

Determinants and Singular Matrices

Ages 14–15

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Determinants and Singular Matrices

Start with the idea, not the formula. A determinant is one number calculated from a square matrix. It gives useful information about that matrix: for a 2 × 2 matrix it tells us whether the matrix can be reversed and, later, how it changes area. A square matrix has the same number of rows and columns, such as 2 × 2. The word diagonal means a line of entries from a corner to the opposite corner. For A = [[a, b], [c, d]], the two diagonal products are a×d and b×c. The 2 × 2 determinant rule is det(A) = (a×d) − (b×c). The symbol det(A) simply means ‘the determinant of A’. If the answer is 0, we call the matrix singular; that means it cannot be reversed. If the answer is not 0, it is non-singular and its inverse may exist.

Your goals

  • ✓You can find a 2 × 2 determinant and use it to test whether a matrix is singular.
  • ✓You can use square matrix, determinant, diagonal, singular, non-singular correctly.
  • ✓You can confirm the matrix is square, multiply the two diagonal pairs, subtract the second product, and interpret zero correctly.

Words to know

square matrix:
A matrix with the same number of rows and columns.
determinant:
A single number calculated from a square matrix.
diagonal:
A line of entries running from one corner of a square matrix to the opposite corner.
singular:
A square matrix whose determinant is zero and which has no inverse.
non-singular:
A square matrix with a non-zero determinant.

Examples

  • 1Step 1 — label the entries: A = 3254, so a = 3, b = 2, c = 5 and d = 4.
  • 2Step 2 — multiply the first diagonal: a×d = 3×4 = 12. Multiply the other diagonal: b×c = 2×5 = 10.
  • 3Step 3 — subtract: det(A) = 12 − 10 = 2. Because 2 is not 0, A is non-singular.
  • 4A zero example: B = 2613. The products are 2×3 = 6 and 6×1 = 6, so det(B) = 6 − 6 = 0. B is singular and has no inverse.
  • 5A negative answer is allowed: det(1432) = 1×2 − 4×3 = 2 − 12 = −10. The minus sign is part of the answer.
  • 6Do not use this 2 × 2 rule on a non-square matrix. A 2 × 3 matrix has no single 2 × 2 determinant.

More than one way to solve it

Try each method, then choose the one you can explain and check most clearly.

Method 1

Two diagonal products

Best for: A 2 × 2 determinant
  1. 1.Check that the matrix has two rows and two columns.
  2. 2.Multiply top-left × bottom-right, then top-right × bottom-left.
  3. 3.Subtract the second product from the first.

Example: [[3, 2], [5, 4]] gives (3×4) − (2×5) = 12 − 10 = 2.

Method 2

Singularity test

Best for: Deciding whether an inverse exists
  1. 1.Find the determinant.
  2. 2.If it is zero, state singular.
  3. 3.If it is non-zero, an inverse is possible.

Example: det([[2, 6], [1, 3]]) = 0, so the matrix is singular.

Check these

  • • Do not add the diagonal products, forget the second diagonal, or calculate a determinant for a non-square matrix.
  • • Keep every number, symbol, label, and unit in its correct place.
  • • Choose the method that makes each step easiest to explain and check.

Helpful habits

  • • Read the whole question before choosing what to do.
  • • Show one clear step at a time and check each step.
  • • If you are stuck, draw a model, try smaller numbers, or revisit the example.

Now try your own

Create your own determinants and singular matrices example, solve it, and explain how you checked it.

Spark

Spark

Creativity & Innovation

“Make up a story about the numbers you see.”