Invertible Matrices and Determinants: Principles

 # Relationship Between Matrix Operations and Inverses

• Contextual Recap of Chapter Two     - In previous sessions, multiplication and addition were defined, alongside various properties like the transpose.     - Multiplication was defined specifically in terms of the dot product, necessitating a detour into Chapter 4 (vectors in $\mathbb{R}^n$) and Section 5.1 (the dot product itself) to deepen the understanding of rows and columns.

• Arithmetic Analogies in Addition     - Addition involves three key components: $A + B$, the additive identity ($O$, the zero matrix), and the additive inverse ($-A$).     - Fundamental relationships: $A + O = O + A = A$ and $A + (-A) = -A + A = O$.     - Note on Universality: The additive inverse ($-A$) exists for all matrices $A$, regardless of whether they are square or rectangular.     - Note on Uniqueness: The zero matrix ($O$) is not unique across the board; its size must match the matrix $A$ it is paired with.

• Extending Analogies to Multiplication     - Multiplicative Identity ($I$): Multiplication is size-sensitive. If $A$ is an $n \times m$ matrix, the identity depends on the side of multiplication:         - Right side: $A \times I_m = A$ (Requires an $m \times m$ identity).         - Left side: $I_n \times A = A$ (Requires an $n \times n$ identity).     - Multiplicative Inverse ($A^{-1}$): We seek a matrix such that combining it with $A$ yields the identity matrix $I$.

Defining and Validating Inverse Matrices

• Theoretical Questions Regarding Inverses     - For a given matrix $A$, are there matrices $B$ and $C$ such that $BA = I$ and $AC = I$?     - Non-commutativity in multiplication suggests no immediate reason that $B$ must equal $C$ or that the resulting identities must be the same size.

• Existence of Inverses     - Inverses for all $A$: False. The zero matrix ($O$) cannot have an inverse because any matrix multiplied by $O$ results in $O$, never $I$.     - Inverses for some $A$: True. For example, if $A = I$, then $I \times I = I$.     - Inverses for none: False (proven by the existence of $I$).

• Proof of Uniqueness ($B = C$)     - Assume $BA = I$ and $AC = I$.     - Start with $BA = I$ and multiply both sides by $C$ on the right: $(BA)C = IC$.     - Using the associative property: $B(AC) = C$.     - Since $AC = I$, then $BI = C$.     - Conclusion: $B = C$. This means for any invertible matrix, the left inverse and right inverse are identical, denoted as $A^{-1}$.

• Notation and Requirements     - The notation is $A^{-1}$, read as "A inverse."     - In order for $A A^{-1} = A^{-1} A = I$ to hold, the matrix $A$ must be square ($n \times n$).

Methods for Finding Inverses

• System of Equations Method (Inefficient)     - Example: Given $A = \begin{pmatrix} 3 & -1 \ 2 & 1 \end{pmatrix}$, find $A^{-1} = \begin{pmatrix} a & b \ c & d \end{pmatrix}$.     - $A A^{-1} = \begin{pmatrix} 3a-c & 3b-d \ 2a+c & 2b+d \end{pmatrix} = \begin{pmatrix} 1 & 0 \ 0 & 1 \end{pmatrix}$.     - This results in two separate $2 \times 2$ systems of equations:         1. $3a - c = 1$ and $2a + c = 0$.         2. $3b - d = 0$ and $2b + d = 1$.     - This can be solved via a $4 \times 4$ checkboard augmented matrix, but it is inefficient due to numerous zeros.

• The Augmented Matrix Method ($[A | I]$)     - Based on the system $Ax = b$. If $A^{-1}$ exists, multiplying both sides on the left yields: $A^{-1}Ax = A^{-1}b$, resulting in $Ix = A^{-1}b$ or $x = A^{-1}b$.     - Process: Augment matrix $A$ with the identity $I$: $[A | I]$.     - Perform row operations until the left side is transformed into $I$.     - The right side will simultaneously transform into $A^{-1}$.     - Final form: $[I | A^{-1}]$.

• Elementary Matrices     - Every row operation can be represented as multiplication by an "Elementary Matrix" ($E$).     - If $k$ operations are required to transform $A$ into $I$, then $E_k \dots E_2 E_1 A = I$.     - Consequently, $E_k \dots E_2 E_1 I = A^{-1}$.

• Example: Finding the inverse of a $3 \times 3$ matrix     - Matrix $A = \begin{pmatrix} 1 & 2 & 0 \ -1 & 1 & 1 \ 0 & 0 & 1 \end{pmatrix}$.     - Set up augmented matrix: $\begin{pmatrix} 1 & 2 & 0 & | & 1 & 0 & 0 \ -1 & 1 & 1 & | & 0 & 1 & 0 \ 0 & 0 & 1 & | & 0 & 0 & 1 \end{pmatrix}$.     - Operation: $R_1 + R_2 \rightarrow R_2$ yields $\begin{pmatrix} 1 & 2 & 0 & | & 1 & 0 & 0 \ 0 & 3 & 1 & | & 1 & 1 & 0 \ 0 & 0 & 1 & | & 0 & 0 & 1 \end{pmatrix}$.     - Operation: $R_2 - R_3 \rightarrow R_2$ yields $\begin{pmatrix} 1 & 2 & 0 & | & 1 & 0 & 0 \ 0 & 3 & 0 & | & 1 & 1 & -1 \ 0 & 0 & 1 & | & 0 & 0 & 1 \end{pmatrix}$.     - Operation: $3R_1 - 2R_2 \rightarrow R_1$ yields $\begin{pmatrix} 3 & 0 & 0 & | & 1 & -2 & 2 \ 0 & 3 & 0 & | & 1 & 1 & -1 \ 0 & 0 & 1 & | & 0 & 0 & 1 \end{pmatrix}$.     - Final Scaling: $\frac{1}{3}R_1$ and $\frac{1}{3}R_2$ yields $\begin{pmatrix} 1 & 0 & 0 & | & 1/3 & -2/3 & 2/3 \ 0 & 1 & 0 & | & 1/3 & 1/3 & -1/3 \ 0 & 0 & 1 & | & 0 & 0 & 1 \end{pmatrix}$.     - Resulting $A^{-1} = \begin{pmatrix} 1/3 & -2/3 & 2/3 \ 1/3 & 1/3 & -1/3 \ 0 & 0 & 1 \end{pmatrix}$.

Theorems and Properties of Inverses

• Invertible Matrix Facts     - If $A$ and $B$ are invertible matrices, $k$ is a positive integer, and $c$ is a nonzero scalar:         - $A^{-1}$, $A^k$, $cA$, and $A^T$ (transpose) are all invertible.         - $(A^{-1})^{-1} = A$         - $(cA)^{-1} = \frac{1}{c} A^{-1}$         - $(A^k)^{-1} = (A^{-1})^k = A^{-k}$         - $(AB)^{-1} = B^{-1} A^{-1}$ (Note the reversed order).         - $(A^T)^{-1} = (A^{-1})^T$

• Elementary Matrices     - All elementary matrices ($E$) are invertible because row operations are reversible.

The Fundamental Theorem of Invertible Matrices (Equivalent Conditions)

• Core Concept     - Let $A$ be an $n \times n$ matrix. The following statements are equivalent (if one is true, all are; if one is false, all are):         1. $A$ is an invertible matrix.         2. $Ax = b$ has a unique solution for all vectors $b$.         3. The homogeneous system $Ax = 0$ has only the trivial solution ($x = 0$).         4. $A$ is row equivalent to the identity matrix $I$.         5. $A$ can be written as a product of elementary matrices.         6. The determinant of $A$ is non-zero ($\det(A) \neq 0$).

Chapter Three: Determinants

• Definition and Domain     - Determinant is a function $\det(A): \text{Set of all } n \times n \text{ matrices} \rightarrow \mathbb{R}$.     - Result of a determinant is always a real number.     - Notation: $\det(A)$ or vertical bars $|A|$. Brackets $[ ]$ denote a matrix, while vertical bars $| |$ denote a determinant scalar.

• Specific Cases for Computation     - 2x2 Determinant:         - $\det \begin{pmatrix} a & b \ c & d \end{pmatrix} = ad - bc$.         - Example: $\det \begin{pmatrix} 2 & 5 \ -3 & 7 \end{pmatrix} = (2 \times 7) - (5 \times -3) = 14 + 15 = 29$.     - 3x3 Determinant (The Basket Technique/Basket Weaving):         - Copy the first two columns to the right of the matrix.         - Sum the products of the diagonals going down from left to right.         - Subtract the sum of the products of the diagonals going up from left to right.         - Warning: This trick only works for $3 \times 3$ matrices.

• Computational Approaches     - Permutation Formula: Inefficient for manual calculation; used for theoretical motivation.     - Cofactor Expansion:         - Can expand about any row or column.         - Generally iterative ($4 \times 4$ matrix breaks into four $3 \times 3$ matrices).         - Scaling: Grows factorially ($O(n!)$). A $55 \times 55$ matrix would take longer than the age of the universe to compute via cofactor expansion on a supercomputer.         - Strategic Tip: Pick the row or column with the most zeros to minimize work.     - Row/Column Reduction: Efficient and scales well. Determinants allow both row and column operations (unlike standard matrices).

Properties of Determinant Operations

• Cofactor Expansion Signs     - Follows a checkerboard pattern:         $\begin{pmatrix} + & - & + \ - & + & - \ + & - & + \end{pmatrix}$.

• Triangular Matrices     - The determinant of an upper or lower triangular matrix is simply the product of its diagonal entries.

• Row/Column Operations and Their Impact on Determinants     1. Swapping: Swapping any two rows or columns changes the sign of the determinant.     2. Scaling Matrix-wide: $\det(cA) = c^n \det(A)$.     3. Scaling a Single Row/Column: If a row or column is multiplied by scalar $c$, the determinant is multiplied by $c$.     4. Row Addition: Replacing a row with the sum of itself and a multiple of another row ($R_2 \rightarrow R_2 + cR_1$) does not change the determinant value.         - Caution: If the row being changed is multiplied by a scalar (e.g., $R_2 \rightarrow 2R_2 + R_1$), the determinant changes and must be adjusted (scaled by $1/c$).

• Invertibility Connection     - A matrix is invertible if and only if $\det(A) \neq 0$.     - If a matrix has a row of zeros in row echelon form, its determinant is zero, and it is therefore not invertible.

Questions & Discussion

• Student Question: "Can we take a break, professor?"

• Instructor Response: "Yeah. We'll take a break then. So why don't we come back at 07:45, and then we'll get back to work on chapter chapter three."