Several apparently different questions about a square matrix always have the same answer. Learning them as one fact saves enormous effort.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
The invertible side
A has an inverse. Its determinant is non-zero. Its columns are independent. Its columns span the whole space.
And also
Ax = b has exactly one solution for every b. Ax = 0 has only the zero solution. Every column has a pivot.
The other side
If any one of those fails, all of them fail. There is no matrix with a zero determinant that still has independent columns.
Why they agree
Each statement describes the same event: whether the transformation flattened space or not. Flattening loses information permanently.
How to use it
Check whichever condition is cheapest. For a small matrix the determinant is usually the fastest door in.
Many statements, one condition
For a square matrix, being invertible, having nonzero determinant, having independent columns, having full rank, and Ax = b always being solvable are all the same condition in different language.
Why the theorem is useful
Whichever version is easiest to check settles all the others. Establishing independence proves solvability without solving anything, which is a genuine economy.
It applies to square matrices only
A non-square matrix has no inverse and no determinant. The equivalences are a theorem about square matrices specifically, and applying them elsewhere is a category error.
Nearly singular is a practical problem
A determinant close to zero means the matrix is almost non-invertible, and solutions become extremely sensitive to small errors. In computation, near-singularity matters more than exact singularity.
Step 2: Try It Yourself
Tap and try it out.
- i-hat lands on(2, 4)
- j-hat lands on(1, 2)
- Determinant0
The parallelogram has collapsed to a segment. The determinant is zero, the whole plane has been squashed onto a line, and no inverse can undo it.
Step 3: Watch an Example
One step at a time.
Watch Omar Settle Four Questions at Once
Omar is asked whether a matrix with rows [2, 4] and [1, 2] is invertible, has independent columns, spans the plane, and gives unique solutions.
- Step 1
He computes the determinant: 4 − 4 = 0.
Step 4: Your Turn
Practice makes it stick.
The One Check
Problem 1 of 2
A square matrix has determinant 0. How many solutions does Ax = 0 have beyond the zero vector? Enter 0 for none, 1 for infinitely many.
The Rank
Problem 2 of 2
An invertible 4 by 4 matrix. How many pivots does it have?
One Fact, Many Faces
1 of 8
Determinant 5. Is the matrix invertible? 1 yes, 0 no.
2 of 8
Determinant 0. Are the columns independent? 1 yes, 0 no.
3 of 8
An invertible 3 by 3 matrix. How many pivots?
4 of 8
Ax = 0 has only the zero solution. Is A invertible? 1 yes, 0 no.
5 of 8
Rows [3, 6] and [1, 2]. Is the matrix invertible? 1 yes, 0 no.
6 of 8
The columns span the whole space. Is the determinant non-zero? 1 yes, 0 no.
7 of 8
Sort each statement by which side of the theorem it belongs to.
Tap something to move it.
- Empty
- Empty
8 of 8
Rows [1, 0] and [0, 1]. Is the matrix invertible? 1 yes, 0 no.
Step 5: Quick Check
Show what you know.
Question 1 of 2
Rows [4, 8] and [1, 2]. Is the matrix invertible? 1 yes, 0 no.
Question 2 of 2
Why do all these conditions agree?
What You Learned
- For a square matrix, invertibility, a non-zero determinant, independent columns and unique solutions are all one fact.
- If one condition fails, every one of them fails.
- They agree because they all describe whether the transformation flattened space.