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Lesson 33 — Transpose, Identity, Inverse Matrix

The transpose mirrors the matrix. The inverse undoes multiplication — exists only when the determinant is non-zero.

Used in: 1.º ano do EM (16 anos) · Math I japonês cap. matrizes · Klasse 11 alemã Lineare Algebra

A A^{-1} = A^{-1} A = I, \qquad (A^T)_{ij} = a_{ji}

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Rigorous notation, full derivation, hypotheses

Transpose and inverse

Transpose

$(A^T)_{ij} = a_{ji}$ . Rows are swapped for columns. Properties:

Property	Formula
Involution	$(A^T)^T = A$
Sum	$(A + B)^T = A^T + B^T$
Scalar	$(\alpha A)^T = \alpha A^T$
Product	$(AB)^T = B^T A^T$ (reverses order)
Inverse-transpose	$(A^T)^{-1} = (A^{-1})^T$

Symmetric matrix: $A^T = A$ . Orthogonal matrix: $A^T A = I$ ⟺ $A^{-1} = A^T$ .

Identity

$I_n$ : $n \times n$ square matrix with 1's on the diagonal and 0 off. For every $A_{n \times n}$ : $AI = IA = A$

Inverse

$A_{n \times n}$ is invertible (non-singular) if there exists $A^{-1}$ such that $AA^{-1} = A^{-1}A = I$ . Equivalence theorem — for square $A$ , the following are equivalent:

$A$ is invertible.
$\det A \neq 0$ .
$A\mathbf{x} = \mathbf{0}$ has only $\mathbf{x} = \mathbf{0}$ .
The columns of $A$ are linearly independent.
The rows of $A$ are linearly independent.
$A$ has full rank: $\text{rank}(A) = n$ .
$A$ is a product of elementary matrices.

2x2 inverse

$A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}, \quad A^{-1} = \frac{1}{ad - bc} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}$

(Valid if $ad - bc \neq 0$ .)

Inverse properties

$(A^{-1})^{-1} = A$
$(AB)^{-1} = B^{-1} A^{-1}$ (reverses order!)
$(A^T)^{-1} = (A^{-1})^T$
$(\alpha A)^{-1} = (1/\alpha) A^{-1}$
$\det(A^{-1}) = 1/\det(A)$

Computation via Gauss-Jordan

Form $[A | I]$ → row-reduce until $[I | A^{-1}]$ . Cost $O(n^3)$ .

Exercise list

48 exercises · 12 with worked solution (25%)

Application 30Understanding 8Modeling 6Challenge 3Proof 1

Ex. 33.1ApplicationAnswer key
Transpose of $\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$ . (Ans: $\begin{pmatrix} 1 & 3 \\ 2 & 4 \end{pmatrix}$ .)
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Ex. 33.2Application
Transpose of $\begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{pmatrix}$ .
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Ex. 33.3ApplicationAnswer key
Inverse of $\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$ . (Ans: $\begin{pmatrix} -2 & 1 \\ 3/2 & -1/2 \end{pmatrix}$ .)
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Ex. 33.4Application
Inverse of $\begin{pmatrix} 2 & 0 \\ 0 & 3 \end{pmatrix}$ .
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Ex. 33.5Application
Inverse of $\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$ .
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Ex. 33.6Application
Does $\begin{pmatrix} 1 & 2 \\ 2 & 4 \end{pmatrix}$ have an inverse? Justify. (Ans: no, $\det = 0$ .)
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Ex. 33.7ApplicationAnswer key
Verify that $A \cdot A^{-1} = I$ for $A = \begin{pmatrix} 3 & 1 \\ 5 & 2 \end{pmatrix}$ .
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Ex. 33.8Application
Inverse of $\begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}$ . (Ans: rotation by $-\theta$ .)
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Ex. 33.9Application
Solve $A\mathbf{x} = \mathbf{b}$ via the inverse: $A = \begin{pmatrix} 2 & 1 \\ 1 & 3 \end{pmatrix}$ , $\mathbf{b} = (5, 7)^T$ .
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Ex. 33.10ApplicationAnswer key
Verify whether $\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$ is symmetric. (Ans: no — $a_{12} \neq a_{21}$ .)
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Ex. 33.11Application
Verify $(AB)^T = B^T A^T$ for $A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$ and $B = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$ .
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Ex. 33.12Application
For which $k$ does the matrix $\begin{pmatrix} 1 & k \\ 2 & 4 \end{pmatrix}$ fail to have an inverse? (Ans: $k = 2$ .)
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Ex. 33.13Application
Inverse of $\begin{pmatrix} 1 & -1 \\ 1 & 1 \end{pmatrix}$ .
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Ex. 33.14Application
Numerically show that $A + A^T$ is symmetric for $A = \begin{pmatrix} 1 & 4 \\ 2 & 5 \end{pmatrix}$ .
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Ex. 33.15Application
Numerically show that $A - A^T$ is skew-symmetric.
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Ex. 33.16ApplicationAnswer key
$(A^{-1})^{-1} = A$ — verify for $A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$ .
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Ex. 33.17Application
For which diagonal $\begin{pmatrix} a & 0 \\ 0 & b \end{pmatrix}$ is invertible? (Ans: $ab \neq 0$ .)
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Ex. 33.18Application
Inverse of $\begin{pmatrix} a & b \\ 0 & d \end{pmatrix}$ (triangular, $ad \neq 0$ ).
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Ex. 33.19Application
$A = \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}$ . Compute $A^4$ and $A^{-1}$ .
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Ex. 33.20ApplicationAnswer key
Decompose $\begin{pmatrix} 1 & 4 \\ 2 & 5 \end{pmatrix}$ as symmetric + skew-symmetric.
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Ex. 33.21Application
Compute $A^{-1}$ of $A = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 5 \end{pmatrix}$ (diagonal).
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Ex. 33.22Application
Compute $A^{-1}$ of $A = \begin{pmatrix} 1 & 1 & 1 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{pmatrix}$ (triangular).
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Ex. 33.23Application
Apply Gauss-Jordan to $[A|I]$ for $A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$ .
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Ex. 33.24Application
Verify that $\begin{pmatrix} 1 & 2 & 3 \\ 0 & 1 & 4 \\ 5 & 6 & 0 \end{pmatrix}$ has an inverse (compute $\det$ ).
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Ex. 33.25Application
Solve $A\mathbf x = \mathbf b$ with $A = \begin{pmatrix} 1 & 1 \\ 2 & 3 \end{pmatrix}$ , $\mathbf b = (3, 8)^T$ , via $A^{-1}$ .
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Ex. 33.26Application
Inverse of the permutation matrix $P = \begin{pmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{pmatrix}$ . (Ans: $P^T = P^{-1}$ .)
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Ex. 33.27Application
Compute $A^{-1}$ for $A = \begin{pmatrix} 1 & 2 \\ -1 & 1 \end{pmatrix}$ .
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Ex. 33.28ApplicationAnswer key
Use $A^{-1}$ to solve $\begin{cases} x + 2y = 5 \\ -x + y = 1 \end{cases}$ .
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Ex. 33.29ApplicationAnswer key
Verify that if $A^T = A^{-1}$ , then $A^TA = I$ (orthogonal matrix).
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Ex. 33.30Application
Verify whether $\begin{pmatrix} 1/\sqrt 2 & 1/\sqrt 2 \\ -1/\sqrt 2 & 1/\sqrt 2 \end{pmatrix}$ is orthogonal.
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Ex. 33.31Understanding
Show that if $A$ is symmetric and invertible, $A^{-1}$ is also symmetric.
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Ex. 33.32UnderstandingAnswer key
Show that if $A^2 = I$ , then $A = A^{-1}$ .
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Ex. 33.33Understanding
Show that an orthogonal matrix ( $A^T A = I$ ) has $A^{-1} = A^T$ .
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Ex. 33.34Understanding
Show that if $A, B$ invertible, $AB$ is invertible.
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Ex. 33.35Understanding
Show that $(A^T)^{-1} = (A^{-1})^T$ .
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Ex. 33.36Understanding
Show that if $A$ is invertible triangular, $A^{-1}$ is also triangular of the same type.
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Ex. 33.37UnderstandingAnswer key
Show that the product of orthogonal matrices is orthogonal.
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Ex. 33.38Understanding
Show that $\det(A^{-1}) = 1/\det(A)$ .
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Ex. 33.39Modeling
Use the inverse to solve: $\begin{cases} 2x + y = 7 \\ x - 3y = -2 \end{cases}$ .
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Ex. 33.40Modeling
In matrix cryptography (Hill cipher), encrypt the message as a vector $\mathbf{m}$ via $A\mathbf{m}$ . Decrypt = $A^{-1}(A\mathbf{m})$ . Model with $A_{2\times 2}$ .
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Ex. 33.41ModelingAnswer key
In CG, the inverse transformation is fundamental: applying a transformation to the camera is applying the inverse to objects. Explain why.
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Ex. 33.42Modeling
In economics, the Leontief matrix $L$ relates production to demand. Solution: $\mathbf{x} = (I - L)^{-1} \mathbf{d}$ . For $L = \begin{pmatrix} 0.2 & 0.1 \\ 0.3 & 0.4 \end{pmatrix}$ , $\mathbf d = (10, 20)^T$ , compute $\mathbf x$ .
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Ex. 33.43Modeling
Identify whether $\begin{pmatrix} 1 & 2 & 3 \\ 0 & 1 & 4 \\ 0 & 0 & 1 \end{pmatrix}$ is upper-triangular. Is the inverse also upper-triangular?
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Ex. 33.44Modeling
In statistics, regression $\hat\beta = (X^TX)^{-1}X^T\mathbf y$ . Why is row reduction preferable to direct inversion?
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Ex. 33.45Challenge
Find a matrix $A$ with $A^3 = I$ but $A \neq I$ . (Hint: rotation by $120°$ .)
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Ex. 33.46ChallengeAnswer key
Show that if $AB = I$ for $A, B \in M_n$ , then $BA = I$ (non-trivial).
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Ex. 33.47Challenge
Compute $A^{-1}$ for $A = I + \mathbf u \mathbf v^T$ with $\mathbf u, \mathbf v \in \mathbb R^n$ and $1 + \mathbf v^T\mathbf u \neq 0$ (Sherman-Morrison).
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Ex. 33.48Proof
Prove $(AB)^{-1} = B^{-1}A^{-1}$ via $(AB)(B^{-1}A^{-1}) = I$ .

Sources

Linear Algebra Done Right — Sheldon Axler · 2024, 4th ed · EN · CC-BY-NC · ch. 3, 7. Primary source.
A First Course in Linear Algebra — Robert A. Beezer · 2022 · EN · GFDL · ch. MISLE.
Álgebra linear — Wikibooks · alive · PT-BR · CC-BY-SA.