Determinant of a Matrix
The determinant is a scalar value that can be computed from a square matrix. It tells us useful information like whether the matrix is invertible.
For a 2x2 matrix A = [[a, b], [c, d]], the determinant is:
det(A) = ad − bc
For larger matrices (like 3x3), we expand using minors and cofactors or row-reduction.
🔍 Example:
For matrix B = [[1, 2], [3, 4]]
det(B) = (1×4) − (2×3) = 4 − 6 = -2
✅ So the matrix is invertible (det ≠ 0).
💡 A matrix is invertible if and only if its determinant is not zero.
Rank of a Matrix
The rank of a matrix is the number of linearly independent rows or columns.
It tells us the dimension of the row space (or column space) of the matrix. The rank is found by reducing the matrix to row echelon form (REF) or reduced row echelon form (RREF).
🔍 Example:
For matrix C = [[1, 2, 3], [2, 4, 6], [3, 6, 9]]
All rows are multiples of the first row, so there’s only one linearly independent row.
✅ Rank = 1
💡 A full-rank matrix has maximum possible rank (equal to the smaller of number of rows or columns).
Nullity of a Matrix
The nullity of a matrix is the number of free variables in the solution of the equation Ax = 0.
In simple terms, it tells us how many directions the solution space "spreads out" in.
You can find nullity using the Rank-Nullity Theorem.
🔍 Rank-Nullity Theorem:
Rank(A) + Nullity(A) = Number of columns in A
🔍 Example:
Let matrix D have 4 columns and Rank(D) = 2
Then:
Nullity(D) = 4 − 2 = 2
✅ So the solution of Dx = 0 has two free variables.
💡 Nullity shows how many solutions (directions) exist in the null space of a matrix.