MAT 242 Chap.4 Inverses of Matrices
Inverses of Matrices
What an inverse is
The inverse of a square matrix A is the matrix that undoes it: multiplying in either order returns the identity. When it exists it is written A⁻¹, and A is called invertible or nonsingular; otherwise A is singular.
Only square matrices qualify, and many square matrices still have no inverse.
Two ways to find it
For 2 × 2 matrices there is a formula worth memorising: swap the diagonal entries, change the sign of the other two, and divide by ad − bc. If that number is zero there is no inverse.
For larger matrices the course uses Gauss-Jordan elimination on the block [A | I]: when the left half becomes I, the right half is A⁻¹; when a row of zeros appears on the left, A is not invertible.
The paid chapter carries a 3 × 3 example through every row operation and then uses the inverse to solve a system as X = A⁻¹B.
Rules that are easy to get backwards
The inverse of a product reverses the order, so the inverse of AB is B⁻¹A⁻¹. An invertible factor can be cancelled only when it sits on the same side of both products.
Division has no meaning for matrices, so the side of the inverse must always be written explicitly.
A grid in the paid chapter lists which statements are valid and why, beside the calculator messages that signal a singular or mis-sized matrix.
The simplest self-check in the whole course lives here: multiply your answer by the original matrix and confirm that the identity comes out.
Linking inverses to the rest of the course
The inverse ties together three ideas that later chapters treat separately.
A square matrix is invertible exactly when its reduced form is the identity, exactly when its determinant is nonzero, and exactly when its rank is full. Coordinates relative to a basis and the change-of-basis matrix are both built from inverses, and so are the least squares formulas near the end of the course.
On Test 1 the inverse is also a solving tool: once A⁻¹ is known, any system with the same coefficients is solved by one multiplication.
Checking an inverse efficiently
A complete check multiplies the proposed inverse by the original matrix and compares the result with the identity.
On a timed test, a partial check is often enough to catch an error: compute only the diagonal entries of the product, which should all be 1, and one or two off-diagonal entries, which should be 0. Any mismatch points to the row of the inverse that needs recomputing.
What this chapter covers
- 01
Invertible and singular matrices
- 02
The 2 × 2 inverse formula
- 03
Inverting with [A | I]
- 04
Solving AX = B with the inverse
- 05
Inverse of a product and cancellation rules
Worked example · free
Invert a 2 × 2 matrix and solve a system with it
- 3ad − bc = 4 − 3 = 1, so the inverse is [2 −3; −1 2].
- 2Multiply the inverse by (4, 1): (8 − 3, −4 + 2) = (5, −2).
- 1Check: 2·5 + 3·(−2) = 4 and 5 + 2·(−2) = 1.
Key terms
- Singular Matrix
- A singular matrix is a square matrix with no inverse, which happens exactly when its determinant is zero.
- Nonsingular Matrix
- A nonsingular matrix is a square matrix that has an inverse; the term means the same as invertible.
- Invertible Matrix
- An invertible matrix is a square matrix A with a matrix A⁻¹ such that AA⁻¹ and A⁻¹A both equal the identity.
Inverses of Matrices FAQ
What is the inverse of a product of two matrices?
It is the product of the two inverses in the opposite order, B⁻¹A⁻¹ for AB. Writing A⁻¹B⁻¹ is a common slip that is only right when the matrices commute.
How do I know the [A | I] method has failed?
If a row of zeros appears in the left block during Gauss-Jordan elimination, the left half can never become the identity, so the matrix is singular and has no inverse.
Can a non-square matrix have an inverse?
No. An inverse must work on both sides, which forces the matrix to be square. Rectangular matrices can still appear in systems, but they are solved by row reduction rather than by inversion.
Exam move
Memorise the 2 × 2 formula until it is automatic, then time one 3 × 3 inversion by [A | I]. Finish every attempt by multiplying back to the identity. Time one full [A | I] inversion per week before Test 1; the procedure is long enough that speed and accuracy both need practice.
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