1 Matrices and Basic Operations
“It is not enough to have a good mind; the main thing is to use it well.”
– René Descartes
1.1 Important Properties
- Commutativity: A\pm B=B\pm A
- Associativity: (A+B)+C=A+(B+C)
- Identity: A+0=0+A=A
- Non-Commutativity: AB\neq BA, in general
- Associativity: (AB)C=A(BC)
- Identity: AI=IA=A
- Zero: A0=0A=0
- Distribution: A(B+C)=AB+AC and (A+B)C=AC+BC
- Associativity: k(A+B)=kA+kB
- Zero: 0A=0
- Idempotent: (A^T)^T=A
- Distribution over Sum: (A+B)^T=A^T+B^T
- Distribution over Product: (AB)^T=B^TA^T
- Idempotent: (A^{-1})^{-1}=A
- Distribution over Product: (AB)^{-1}=B^{-1}A^{-1}
- Transpose: (A^{-1})^T=(A^T)^{-1}
- Distribution over Product: det(AB)=det(A) det(B)
1.2 Exercises
Exercise 1.1 Solve the system by hand: \begin{aligned} x + y + z &= -1 \\ 5x + 3y - z &= 9 \\ 3x + 2y - 2z &= 5 \end{aligned}
Exercise 1.2 Can two matrices be always added?
Exercise 1.3 If addition is possible, add the following matrices: A=\begin{bmatrix} -1 & 2 & 3\\ 0 & 1 & 5 \end{bmatrix} \text{ and } B=\begin{bmatrix} 1/2 & 1 & 3\\ 2 & -2 & 0 \end{bmatrix}.
Exercise 1.4 For matrices A,B, is it always true that A+B=B+A? Explain your answer.
Exercise 1.5 If multiplication is possible, multiply the following matrices: A=\begin{bmatrix} 1 & 2 & 3\\ 0 & -1 & 5 \end{bmatrix} \text{ and } B=\begin{bmatrix} 1 & 2 & 1\\ 1 & 0 & 2 \end{bmatrix}.
Exercise 1.6 If multiplication is possible, multiply the following matrices: A=\begin{bmatrix} 1 & 2\\ 0 & -1\\ -1 & 3 \end{bmatrix} \text{ and } B=\begin{bmatrix} 1 & 5 & 1\\ 1 & 12 & 2 \end{bmatrix}.
Exercise 1.7 For matrices A and B, is it always true that AB=BA? Explain your answer.
Exercise 1.8 When do the matrix products AB and BA have the same size? Explain your answer.
Exercise 1.9 Give an example of a 4\times 2 diagonal matrix.
Exercise 1.10
- If D=\begin{bmatrix} a & 0 \\ 0 & b \\ \end{bmatrix}
is a 2 \times 2 diagonal matrix with a=b, then characterize the only matrices that commute (under matrix multiplication) with D.
- What if a=b ?
Exercise 1.11
Is the sum of two diagonal matrices of the same size a diagonal matrix ?
Is the product of two diagonal matrices of the same size a diagonal matrix ?
Exercise 1.12 Intuitively justify the following fact: (AB)^T=B^TA^T.
Exercise 1.13 Is the sum of two symmetric matrices always symmetric?
Exercise 1.14 Is the product of two symmetric matrices always symmetric?
Exercise 1.15 Give an example of a skew-symmetric matrix of order 4\times 4 with at least 4 non-zero elements.
Exercise 1.16 For any matrix A, explain why
A+A^T is symmetric.
A-A^T is skew-symmetric.
Exercise 1.17 If A has an inverse, then show that the inverse is unique.
Exercise 1.18 If A is the inverse of B, then show that B is the inverse of A.
Exercise 1.19
- Is \text{det} (A+B)=\text{det} (A)+ \text{det} (B)?
- Is \text{det} (AB)=\text{det} (A) \text{det} (B)?
Exercise 1.20 Suppose A is a 2 \times 2 matrix and B is the matrix we get by interchanging the two rows of A. Show that \text{det} B = - \text{det} A.
Exercise 1.21 Find the determinant of \begin{bmatrix}-1 & 2 & 3\\ 0 & 1 & 5\\ 4 & 1 & -1\end{bmatrix}.
Exercise 1.22 Find a 3\times 3 matrix whose determinant is 6 and the absolute value of each entry is no bigger than 3.
Exercise 1.23 \begin{bmatrix} 2 & 1 & 3 \end{bmatrix}\begin{bmatrix} -1 & 0 & -1\\ -1 & 1 & 0\\ 0 & 1 & 1 \end{bmatrix}\begin{bmatrix} 1 \\ 0 \\ -1 \end{bmatrix}=?
Exercise 1.24 Find the inverse of the following matrix: A=\begin{bmatrix} 1 & 2 & 3\\ 0 & -1 & 2\\ 1 & -2 & 0 \end{bmatrix}.
Exercise 1.25 Find the inverse of the following matrix: \begin{bmatrix} -1 & 0 & -1\\ -1 & 1 & 0\\ 0 & 1 & 1 \end{bmatrix}.
Exercise 1.26 If =\begin{bmatrix} 5 & 3\\ -1 & -2\\ \end{bmatrix}, then show that A satisfies the equation A^2-3A-7I=0.
Exercise 1.27 Consider a n \times n matrix A. Then:
- If A is invertible and AB=0 for some n \times n matrix B, then B=0.
- If A is not invertible, then we can find a n \times n matrix B with AB=0 but B \ne 0.
