After reviewing class procedures, and items on the syllabus, we began a discussion of \(2\times 2\) matrices over \(\mathbb{R}\), the set of which we denote by \(\textrm{M}_2(\mathbb{R})\). Given \(A = \begin{pmatrix} a & b\\c & d\end{pmatrix}\), we identified the entries of \(A\) as follows: \(a\) is the (1,1) entry, \(b\) is the (1,2) entry, \(c\) is the (2,1) entry and \(d\) is the (2,2) entry.
We established two fundamental operations:
- (i) Matrix addition: Given \(A = \begin{pmatrix} a & b\\c & d\end{pmatrix}\), \(B = \begin{pmatrix} e & f\\g & h\end{pmatrix}\), define \(A+B := \begin{pmatrix} a+e & b+f\\c+g & d+h\end{pmatrix}\).
- (ii) Scalar multiplication: Given \(\lambda \in \mathbb{R}\) and \(A\in \textrm{M}_2(\mathbb{R})\), define \(\lambda \cdot A = \begin{pmatrix} \lambda a & \lambda b\\\lambda c & \lambda d\end{pmatrix}\).
We then discussed at length the following properties. In what follows, \(A, B, C\in \textrm{M}_2(\mathbb{R})\) and \(\lambda, \lambda_1, \lambda_2 \in \mathbb{R}\).
1. An additive identity exists: For \({\bf 0}_{2\times 2} := \begin{pmatrix} 0 & 0\\0 & 0\end{pmatrix}\), we have \({\bf 0}_{2\times 2}+A = A\), for all \(A\in \textrm{M}_2(\mathbb{R})\).
2. Additive inverses exist: Given \(-A := \begin{pmatrix} -a & -b\\-c & -d\end{pmatrix}\), we have \(-A+A = {\bf 0}_{2\times 2}\).
3. Addition is commutative: \(A+B = B+A\).
4. Addition is associative: \((A+B)+C = A+(B+C)\), for \(A, B, C\in \textrm{M}_2(\mathbb{R})\).
5. Scalar multiplication distributes over matrix addition: \(\lambda \cdot (A+B) = \lambda \cdot A+\lambda \cdot B\), for \(\lambda \in \mathbb{R}\).
6. Scalar addition is distributive: \((\lambda_1+\lambda_2)\cdot A = \lambda_1\cdot A+\lambda_2\cdot A\), for \(\lambda_1,\lambda_1\in \mathbb{R}\).
7. Scalar multiplication is associative: \((\lambda_1\lambda_2)\cdot A = \lambda _1\cdot (\lambda_2\cdot A)\).
8. \(1\cdot A = A\) and \(0\cdot A = {\bf 0}_{2\times 2}\).
We also discussed how one might prove these identities and noted that the properties above will be recurring throughout the semester as we discuss abstract vector spaces. We ended class by discussing the following consequences of properties (1)-(8) above. Keeping the same notation, we have
- (i) \(-1\cdot A = -A\).
- (ii) Additive inverses are unique, i.e., if \(A+C = {\bf 0}_{2\times 2}\), then \(C = -A\).
- (iii) Cancellation holds for matrix addition, i.e., if \(A+B = A+C\), then \(B = C\).
In today's lecture we introduced two new operations for \(2\times 2\) matrices, namely multiplication of a matrix times a column and multiplication of two (\(2\times 2\)) matrices.
For \(A = \begin{pmatrix} a & b\\c & d\end{pmatrix}\), \(B = \begin{pmatrix} e & f\\g & h\end{pmatrix}\), \(C = \begin{pmatrix} u\\v\end{pmatrix}\), we defined
- (i) \(A\cdot C = \begin{pmatrix} a & b\\c & d\end{pmatrix} \cdot \begin{pmatrix} u\\v\end{pmatrix} := \begin{pmatrix} au+bv\\cu+dv\end{pmatrix}\).
- (ii) \(A\cdot B = \begin{pmatrix} a & b\\c & d\end{pmatrix}\cdot \begin{pmatrix} e & f\\g & h\end{pmatrix} = \begin{pmatrix} ae+bg & af+bh\\ce+dg & cf+dh\end{pmatrix}\).
We noted that the (1,1) entry of \(AB\) is \(R_1\cdot C_1\), the (1,2) entry is \(R_1\cdot C_2\), the (2,1) entry is \(R_2\cdot C_1\) and the (2,2) entry is \(R_2\cdot C_2\), where \(R_i\) is the \(i\)th row of \(A\) and \(C_j\) is the \(j\)th column of \(B\). We also noted that if we think of \(B\) as the matrix with columns \(C_1, C_2\), i.e., \(B = [C_1\ C_2]\), then \(AB = [AC_1\ AC_2]\), the matrix with columns \(AC_1, AC_2\).
We discussed how we can use the product of a matrix times a column to re-write a system of equations as a single matrix equation, as follows. Given the system of two equations in two unknowns
We can write this as \(AX = L\), where \(A = \begin{pmatrix} a & b\\c & d\end{pmatrix}\), \(X = \begin{pmatrix} x\\y\end{pmatrix}\) and \(L = \begin{pmatrix} u\\v\end{pmatrix}\).
We then discussed powers of \(2\times 2\) matrices and the class calculated \(A^2, A^3, A^4\) and conjectured the value of \(A^{2026}\), for \(A = \begin{pmatrix} 1 & 0\\1 & 1\end{pmatrix}\). We were easily able to conjecture \(A^n = \begin{pmatrix} 1 & 0\\n & 1\end{pmatrix}\), for all \(n\geq 1\), which led to a discussion of how to use mathematical induction to prove this fact. One first establishes the base case \(n = 1\), which in this case is clear. One then shows that the \(n-1\) case implies the \(n\)th case - the inductive step, which in this case amounted to showing that \(A\cdot \begin{pmatrix} 1 & 0\\n-1 & 1\end{pmatrix} = \begin{pmatrix} 1 & 0\\n & 1\end{pmatrix} = A^n\). The class then used induction to prove the formula \(1+2+\cdots + n = \frac{n(n+1)}{2}\).
We moved on to discuss (but not prove) the following
Properties of matrix multiplication. Let \(A, B, C\) be \(2\times 2\) matrices.
- (i) \({\bf 0}_{2\times 2}\cdot A = {\bf 0}_{2\times 2} = A\cdot {\bf 0}_{2\times 2}\).
- (ii) For \(I_2 := \begin{pmatrix} 1 & 0\\0 & 1\end{pmatrix}\), \(A\cdot I_2 = A = I_2\cdot A\), i.e., a multiplicative identity exists.
- (iii) Multiplication distributes over matrix sums: \(A\cdot (B+C) = A\cdot B+A\cdot C\).
- (iv) Multiplication is associative: \(A(BC) = (AB)C\).
- (v) A matrix \(D\) satisfying \(AD = I_2 = DA\) is called an inverse of \(A\) and is denoted \(A^{-1}\).
We finished class by noting that if the matrix equation \(AX = L\) represents a system of equations (as above) and \(A\) has an inverse, then we can multiply both sides of the matrix equation by \(A^{-1}\) to get the solution \(X = A^{-1}L\).
We began class with the following definition. For the \(2\times 2\) matrix \(A = \begin{pmatrix} a & b\\c & d\end{pmatrix}\), the determinant of \(A\), denoted \(\det A\), equals \(ad-bc\).
We then discussed and verified the following
Properties of the determinant. Let \(A, B\) denote \(2\times 2\) matrices over \(\mathbb{R}\).
- (i) If \(A'\) is obtained from \(A\) by multiplying a row or column of \(A\) by \(\lambda \in \mathbb{R}\), then \(\det A' = \lambda \det A\).
- (ii) If \(A'\) is obtained from \(A\) by interchanging its rows or interchanging its columns, then \(\det A' = -\det A\).
- (iii) If \(A'\) is obtained from \(A\) by adding a multiple of one of its rows to another row, then \(\det A = \det A'\). Similarly for the columns of \(A\).
The operations (i)-(iii) are called elementary row or column operations.
- (iv) \(\det AB = \det A\cdot \det B\).
- (v) Suppose \(\det A \not = 0\). Set \(\Delta := \det A\). Then \(A^{-1}\) exists and we have \(A^{-1} = \begin{pmatrix} \frac{d}{\Delta} & -\frac{c}{\Delta}\\-\frac{b}{\Delta} & \frac{a}{\Delta}\end{pmatrix}\).
- (vi) Given vectors \(u = (a,b)\) and \(v = (c,d)\), the area of the parallelogram in \(\mathbb{R}^2\) spanned by \(u\) and \(v\) is \(|ad-bc|\), i.e., the absolute value of \(\det \begin{pmatrix} a & b\\c & d\end{pmatrix}\).
We ended class by looking at a typical system of two linear equations in two unknowns
We noted that each equation corresponds to a straight line \(L_1, L_2\) (respectively) in \(\mathbb{R}^2\) and \((s,t)\) is a solution to the system if and only if \((s,t)\) is a point on each line. Thus the following are the only possibilities for the solution set to the given system of equations:
- (i) There is a unique solution. This occurs when \(L_1\) and \(L_2\) are not parallel, and thus intersect in a single point.
- (ii) There is no solution. This occurs when \(L_1\) and \(L_2\) are parallel.
- (iii) There are infinitely many solutions. This occurs when \(L_1 = L_2\), so that \((s,t)\) is a solution to the system if and only if it is a solution to the first (or second) equation.
Thus, there can never be a \(2\times 2\) system of linear equations with exactly 17 solutions! (Or with exactly \(n\) solutions for any \(n > 1\).)
In the previous lecture we saw that given a system of linear equations
\[\begin{align*} ax+by &= u\\ cx+dy &= v \end{align*}\]whose coefficient matrix \(A = \begin{pmatrix} a & b\\c & d\end{pmatrix}\) has non-zero determinant \(\Delta\), then the solution to the system is given by
Cramer's Rule. For the system above, with \(\Delta \not = 0\), \(x = \frac{\det\begin{pmatrix} u & b\\v & d\end{pmatrix}}{\Delta}\) and \(y = \frac{\det\begin{pmatrix} a & u\\c & v\end{pmatrix}}{\Delta}\).
We noted that for large systems of linear equations, Cramer's rule is not cost effective, so we began a discussion of Gaussian elimination. We started with a specific system of equations (like)
\[\begin{align*} 2x+6y &= 8\\ 3x+y &= 4 \end{align*}\]and performed a sequence of operations that changed the system, but preserved the solution. These operations were of the following form: Interchange equations, add a multiple of one equation to another equation, and multiply an equation by a non-zero number. This simplified the system to one trivially solvable, namely
\[\begin{align*} x &= 1\\ y &= 1. \end{align*}\]We noted that in doing the various operations, the arithmetic involved the coefficients in the equations and the variables were essentially placeholders. This led to considering the corresponding augmented matrix \(\begin{bmatrix}2 & 6 & | & 8\\3 & 1 & | & 4\end{bmatrix}\). By performing the same operations on the rows of the augmented matrix that we did on the system of equations, this led to the augmented matrix \(\begin{bmatrix}1 & 0 & | & 1\\0 & 1 & | & 1\end{bmatrix}\), which corresponds to the system
\[\begin{align*} x &= 1\\ y &= 1. \end{align*}\]We formalized this process by defining
Elementary Row Operations. Let \(A\) be a \(2\times 2\) matrix (or any matrix in fact). The following constitute elementary row operations:
- (i) Interchange two rows.
- (ii) Add a multiple of one row to another row.
- (iii) Multiply a row by a non-zero number.
We noted that the goal was to put, if possible, the beginning augmented matrix \(\begin{bmatrix}a & b & | & u\\c & d & | & v\end{bmatrix}\) into the form \(\begin{bmatrix}1 & 0 & | & s\\0 & 1 & | & t\end{bmatrix}\), from which the solution \(x = s, y = t\) could be read. We noted that the strategy for the Gaussian elimination process should be as follows: Using elementary row operations, first get a 1 in the (1,1) entry of the augmented matrix, then use that 1 to get 0 below it. Then make, if possible, the (2,2) entry of the augmented matrix 1 and then use that 1 to get a 0 above it. This will always be possible when the original system has a unique solution.
We ended class by considering the remaining two cases. In one case, the final augmented matrix took the form \(\begin{bmatrix}1 & 3 & | & 4\\0 & 0 & | & 0\end{bmatrix}\), which corresponds to the system \(x+3y = 4\), from which one concludes \(x = 4-3y\). To describe the solution set we introduced another parameter \(t\) to get \(\{(4-3t, t)\ |\ t\in \mathbb{R}\}\) as the solution set. This is the case when the system has infinitely many solutions. We then saw an example where the final augmented matrix took the form \(\begin{bmatrix}1 & 3 & | & 4\\0 & 0 & | & 1\end{bmatrix}\), which meant the original system had no solution, since \(0 = 1\) is a contradiction.