A square matrix A is invertible when some matrix A⁻¹ satisfies AA⁻¹ = A⁻¹A = I. To find it, row reduce the augmented matrix [A | I]: if A reduces to the identity, the right half becomes A⁻¹, and if a zero row appears on the left, A has no inverse. The inverse undoes the transformation A performs, solves Ax = b as x = A⁻¹b, and reverses products in the order (AB)⁻¹ = B⁻¹A⁻¹. For square matrices, being invertible is equivalent to a long list of other conditions from the course.
What you'll learn
Find the inverse of a matrix by row reducing [A | I]
Recognize a matrix with no inverse
Solve Ax = b with an inverse
Use (AB)⁻¹ = B⁻¹A⁻¹ and the Invertible Matrix Theorem
A matrix A transforms vectors. If some other matrix transforms them back,
it is the inverse of A:
AA−1=IandA−1A=I
Only square matrices can have two-sided inverses, and many square matrices have
none. The precalculus formula handles the 2×2
case:
[acbd]−1=ad−bc1[d−c−ba]ad−bc=0
For larger matrices, row reduction does the job.
Finding A−1. Row reduce the n×2n matrix [A∣I]. If
the left half becomes I, the right half is A−1. If a row of zeros
appears on the left, A has no inverse.
Every row operation is the same as multiplying on the left by a matrix, called
an elementary matrix: do the operation to I and you have it. A sequence
of row operations that turns A into I is therefore a product
E=Ek⋯E2E1 with EA=I, which says E=A−1. Performing the
same operations on I builds EI=E on the right-hand side. The row
operations that reduce A to I multiply together to make A−1, and
carrying I alongside records that product.
For an n×n matrix A, the following statements are either all true or
all false:
A is invertible.
A row reduces to I; equivalently, it has n pivots.
Ax=0 has only the trivial solution.
The columns of A are linearly independent.
Ax=b has exactly one solution for every b.
The columns of A span ℝⁿ.
The transformation x↦Ax is one-to-one and onto.
Every item is a way of saying ”n pivots in an n×n matrix”. For a
square matrix a pivot in every column and a pivot in every row are the same
condition, which is why independence and spanning coincide here and nowhere
else.
ad−bc=6−5=1. Swap the diagonal entries, negate the others, and divide by 1.
Invert [4276].
Answer
[0.6−0.2−0.70.4]
Full solution
ad−bc=24−14=10, so the inverse is 101[6−2−74].
Solve [2513]x=[12] with the inverse from Example 1.
Answer
x=(1,−1)
Full solution
[3−5−12][12]=[3−2−5+4].
Does [2163] have an inverse?
Answer
No
Full solution
ad−bc=6−6=0. The first row is twice the second, so row reduction leaves a zero row.
Invert 200040005.
Answer
210004100051
Full solution
A diagonal matrix scales each axis, and scaling back takes the reciprocals.
Invert 101111011.
Answer
01−1−1101−11
Full solution
Row reduce [A∣I]: R3−R1 gives (0,0,1∣−1,0,1), which is already the third row of the inverse. Then R2−R3 gives (0,1,0∣1,1,−1), and R1−R2 gives (1,0,0∣0,−1,1).
If A and B are invertible 3×3 matrices, write (AB)−1 and (A−1)−1.
Answer
B−1A−1 and A
Full solution
(AB)(B−1A−1)=A(BB−1)A−1=AA−1=I. And A undoes A−1, so it is its inverse.
A 4×4 matrix has independent columns. Is it invertible?
Answer
Yes
Full solution
Independent columns give four pivots, and by the Invertible Matrix Theorem a square matrix with n pivots is invertible.
For a 5×5 matrix, Ax=0 has a nonzero solution. Can Ax=b have a unique solution for some b?
Answer
No
Full solution
A nonzero solution of the homogeneous system means a free variable, so every consistent system Ax=b has infinitely many solutions.
A student inverts [1324] by taking reciprocals of the entries. What went wrong?
Hint
Multiply the student’s matrix by the original.
Answer
Reciprocals do not give the inverse. The inverse is [−21.51−0.5].
Full solution
The product of the original with the matrix of reciprocals has first entry 1⋅1+2⋅31=35, not 1. With ad−bc=4−6=−2, the formula gives −21[4−3−21].
Frequently asked questions
How do you find the inverse of a matrix?
Row reduce the augmented matrix [A | I]. When the left half becomes I, the right half is A⁻¹.
How can you tell that a matrix has no inverse?
Row reduction of A produces a row of zeros, meaning fewer than n pivots. Such a matrix is called singular.
What is the inverse of a product?
(AB)⁻¹ = B⁻¹A⁻¹. The order reverses, the way taking off shoes and socks reverses putting them on.
Do non-square matrices have inverses?
Not two-sided ones. An inverse must undo the transformation in both directions, which needs the input and output spaces to have the same dimension.
What is the Invertible Matrix Theorem?
A list of conditions on a square matrix that are all true or all false together: invertible, n pivots, independent columns, columns spanning ℝⁿ, Ax = 0 having only the trivial solution, and more.