Matrix determinant lemma

Key takeaways

Matrix determinant lemma belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Matrix determinant lemma to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Matrix determinant lemma from memory before moving on to harder problems.

Reference excerpt

In mathematics, in particular linear algebra, the matrix determinant lemma computes the determinant of the sum of an invertible matrix A and the dyadic product, u vT, of a column vector u and a row vector vT.

Statement Suppose A is an invertible square matrix and u, v are column vectors. Then the matrix determinant lemma states that

det ( A + u v T ) = ( 1 + v T A − 1 u ) det ( A ) . {\displaystyle \det(\mathbf {A} +\mathbf {uv} ^{\textsf {T}})=(1+\mathbf {v} ^{\textsf {T}}\mathbf {A} ^{-1}\mathbf {u} )\,\det(\mathbf {A} )\,.}

Here, uvT is the outer product of two vectors u and v. The theorem can also be stated in terms of the adjugate matrix of A:

det ( A + u v T ) = det ( A ) + v T a d j ( A ) u , {\displaystyle \det(\mathbf {A} +\mathbf {uv} ^{\textsf {T}})=\det(\mathbf {A} )+\mathbf {v} ^{\textsf {T}}\mathrm {adj} (\mathbf {A} )\mathbf {u} \,,}

in which case it applies whether or not the matrix A is invertible.

Proof First the proof of the special case A = I follows from the equality:

( I 0 v T 1 ) ( I + u v T u 0 1 ) ( I 0 − v T 1 ) = ( I u 0 1 + v T u ) . {\displaystyle {\begin{pmatrix}\mathbf {I} &0\\\mathbf {v} ^{\textsf {T}}&1\end{pmatrix}}{\begin{pmatrix}\mathbf {I} +\mathbf {uv} ^{\textsf {T}}&\mathbf {u} \\0&1\end{pmatrix}}{\begin{pmatrix}\mathbf {I} &0\\-\mathbf {v} ^{\textsf {T}}&1\end{pmatrix}}={\begin{pmatrix}\mathbf {I} &\mathbf {u} \\0&1+\mathbf {v} ^{\textsf {T}}\mathbf {u} \end{pmatrix}}.}

The determinant of the left hand side is the product of the determinants of the three matrices. Since the first and third matrix are triangular matrices with unit diagonal, their determinants are just 1. The determinant of the middle matrix is our desired value. The determinant of the right hand side is simply (1 + vTu). So we have the result:

det ( I + u v T ) = 1 + v T u . {\displaystyle \det(\mathbf {I} +\mathbf {uv} ^{\textsf {T}})=1+\mathbf {v} ^{\textsf {T}}\mathbf {u} .}

Then the general case can be found by setting u to be A−1u:

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Matrix determinant lemma

Start with the simplest possible case. Write down what Matrix determinant lemma claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Matrix determinant lemma before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Matrix determinant lemma ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Matrix determinant lemma

In research: Matrix determinant lemma appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Matrix determinant lemma in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Matrix determinant lemma is common in secondary-school and first-year university syllabi. It links to neighbouring topics Lemmas in linear algebra, Matrix theory, so understanding it makes those chapters shorter.
In everyday life: Look for Matrix determinant lemma outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Matrix determinant lemma in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Matrix determinant lemma means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Matrix determinant lemma out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Matrix determinant lemma in simple terms?

In mathematics, in particular linear algebra, the matrix determinant lemma computes the determinant of the sum of an invertible matrix A and the dyadic product, u vT, of a column vector u and a row vector vT. Statement Suppose A is an invertible square matrix and u, v are column vectors.

Why does Matrix determinant lemma matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Matrix determinant lemma?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Matrix determinant lemma.