Introductory Linear Algebra/Matrix inverses and determinants

Matrix inverses
are analogous to the (or reciprocal) in the number system.

In the number system, the multiplicative inverse, if it exists, is unique. Indeed, the matrix inverse, if it exists, is also unique similarly, as shown in the following proposition.

Matrix inverse can be used to solve SLE, as follows:

Then, we will define the, which is closely related to EROs, and is important for the proof of results related to EROs.

Then, we will state a of invertible matrix theorem, in which some results from the complete version of invertible matrix theorem are removed.

The following provides us a convenient and efficient way to find the inverse of a matrix.

Determinants
Then, we will discuss the, which allows characterizing some properties of a matrix.

For the formula of determinants of $$3\times 3$$ matrices, we have a useful mnemonic device for it, namely the, as follows:

Then, we will given an example about computing the determinant of a $$4\times 4$$ matrix, which cannot be computed by the Rule of Sarrus directly.

Indeed, we can compute a determinant by the along an arbitrary row, as in the following theorem.

Its proof (for the general case) is complicated, and thus is skipped.

Then, we will discuss several properties of determinants that ease its computation.

Then, we will introduce a convenient way to determine invertibility of a matrix. Before introducing the theorem, we have a lemma.

After introducing this result, we will give some properties of determinants which can ease the computation of determinants.

Then, we will introduce of matrix, which has a notable result related to computation of matrix inverse.

Then, we will introduce a result that allows us to compute the unique solution of SLE directly, namely.