2×2 Matrix Rank Calculator

  The three possible ranks of a 2×2 matrix

The rank of a matrix is the number of linearly independent rows, which always equals the number of linearly independent columns. A 2×2 matrix has only three possibilities.

Rank 0: every entry is zero. Only the zero matrix has rank 0.

Rank 1: the matrix is not zero, but one row is a multiple of the other. This is exactly the case where the determinant a·d − b·c equals zero.

Rank 2: the determinant is not zero. The rows are independent, the matrix is invertible and it is said to have full rank.

So for a 2×2 matrix one subtraction settles the question: compute a·d − b·c. If it is not zero the rank is 2. If it is zero the rank is 1, unless all four entries are zero.

  Worked example

Take the matrix with rows (2, 4) and (1, 2). The determinant is 2·2 − 4·1 = 0, so the rank is less than 2. The matrix is not the zero matrix, so the rank is 1. Row reduction shows the same thing: subtracting 1/2 times row 1 from row 2 leaves the rows (2, 4) and (0, 0), with a single non-zero row.

Change one entry to get the rows (2, 4) and (1, 3). Now the determinant is 2·3 − 4·1 = 2, which is not zero, so the rank is 2. Subtracting 1/2 times row 1 from row 2 gives (0, 1), and both rows are non-zero.

  What the rank tells you

For a system of two linear equations in two unknowns, rank 2 of the coefficient matrix means the two lines cross in exactly one point. Rank 1 means the lines are parallel or the same line, so there is no solution or infinitely many.

As a transformation of the plane, a rank 2 matrix maps the plane onto the whole plane, a rank 1 matrix squashes it onto a line, and the zero matrix sends everything to the origin.

The calculator above is set to 2×2 and shows the row operations it uses to reach the answer.