ECON 441: Introduction to Mathematical Economics
Lecture 5
For a square matrix \(A\), its inverse \(A^{-1}\) is defined as: \[A A^{-1}=A^{-1} A=I\]
Squareness is a necessary condition, not a sufficient condition
If a matrix’s inverse exists, it is called a nonsingular matrix
\[\left(A^{-1}\right)^{-1}=A\]
\[(A B)^{-1}=B^{-1} A^{-1}\]
\[\left(A^T\right)^{-1}=\left(A^{-1}\right)^T\]
\[Ax = b\]
Pre-multiply both sides by \(A^{-1}\), \[A^{-1} Ax = A^{-1} b \quad \implies x = A^{-1} b\]
If \(A\) is singular, a unique solution does not exist.
Squareness is necessary but not sufficient
Sufficient condition for nonsingularity:
Rows (or equivalently) columns are linearly independent
Example. \[A=\begin{bmatrix} 1 & 2 \\ 2 & 4 \end{bmatrix} \quad B=\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}\]
\(A\) is singular, \(B\) is nonsingular.
\[A=\begin{bmatrix} 1 & 2 \\ 2 & 4 \end{bmatrix} \quad x=\begin{bmatrix} x_1 \\ x_2 \end{bmatrix} \quad d=\begin{bmatrix} a \\ b \end{bmatrix}\]
We have a system of linear equations: \[Ax = d\]
Then, \[x_1+2x_2 = a\] \[2x_1+4x_2 = b\]
\[x_1+2x_2 = a\] \[2x_1+4x_2 = b\]
For these equations to be consistent, we need \(b=2a\): \[x_1+2x_2 = a\] \[2x_1+4x_2 = 2a\]
Both are the same equation, infinite number of solutions.
To summarize, for a matrix to be nonsingular (i.e. its inverse exists):
Necessary condition: squareness
Sufficient condition: rows or (equivalently) columns are linearly independent
Exercise 4.6: 6 (added to Practice Problems 4)
Exercise 5.1: 1 – necessary and sufficient conditions (in Practice Problems 2)
Textbook reference: 4.6, 5.1
ECON 441 · Introduction to Mathematical Economics