Matrices and Determinants

Matrices carry about two questions: the order of a matrix or a product, a 2 × 2 determinant or inverse, and one property question (singular, symmetric, effect of scalar multiplication).

The Dean's Office model asks "how many elements in a single column of a 5 × 3 matrix". Matrices are written here in rows as [[a, b], [c, d]].

6.1 Types, order and operations on matrices

• A matrix with m rows and n columns has order m\times n and mn elements; each column has m elements.
• Types: row, column, square, diagonal, scalar, identity I, zero (null), symmetric (A=A^T), skew-symmetric (A=-A^T, diagonal zero).
• A+B needs equal order. AB exists only if columns of A = rows of B; order of AB is (rows of A)×(columns of B). In general AB\ne BA; AI=IA=A.
• (A+B)^T=A^T+B^T and (AB)^T=B^TA^T.

(a) Arthur Cayley, theory of matrices (1858)

(b) J. J. Sylvester, who named the matrix

6.2 Determinants, minors and cofactors

• For [[a, b], [c, d]], |A|=ad-bc. For a 3 × 3, expand along any row: a_{11}C_{11}+a_{12}C_{12}+a_{13}C_{13} with cofactor C_{ij}=(-1)^{i+j}M_{ij}.
• A determinant is 0 if two rows (or columns) are equal or proportional or a row is all zero. Swapping two rows changes the sign. A triangular matrix has |A|= product of the diagonal.
• |AB|=|A||B|, |A^T|=|A|, and |kA|=k^n|A| for order n.

6.3 Adjoint, inverse and linear systems

• A^{-1}=\frac{\text{adj}A}{|A|} exists iff |A|\ne0 (non-singular). For 2 × 2 [[a, b], [c, d]]: \text{adj}A= [[d, -b], [-c, a]].
• AA^{-1}=I, (AB)^{-1}=B^{-1}A^{-1}, |A^{-1}|=\frac{1}{|A|}, |\text{adj}A|=|A|^{n-1}.
• Cramer's rule for a_1x+b_1y=c_1, a_2x+b_2y=c_2: x=\frac{D_x}{D}, y=\frac{D_y}{D}, D=a_1b_2-a_2b_1, with D\ne0.

6.4 Eight traps in matrices and determinants

Matrix questions reward a clear habit: check the order, then the sign, then the rule. These eight traps cover the model-paper style questions on products, determinants and inverses.

1. Check the order before you multiply. A_{m\times n}B_{n\times p} is defined only if the columns of A equal the rows of B, and the product is m\times p (Fig 6.1). A 2\times3 times a 3\times2 gives 2\times2, but a 2\times3 times another 2\times3 is undefined. Always write the orders beside the matrices first.
2. AB\neq BA in general. Even when both products exist they are usually different, as PQ and QP show in Fig 6.1. So (A+B)^2=A^2+AB+BA+B^2, not A^2+2AB+B^2. The reversal laws are (AB)^T=B^TA^T and (AB)^{-1}=B^{-1}A^{-1}.
3. A zero product does not mean a zero factor. AB=0 can happen with A\neq0 and B\neq0, for example \begin{pmatrix}1&0\\0&0\end{pmatrix}\begin{pmatrix}0&0\\0&1\end{pmatrix}=0. Likewise AB=AC does not give B=C unless A is invertible. Do not cancel matrices as you cancel numbers.
4. Only square matrices have determinants, and they obey set rules. |kA|=k^n|A| for an n\times n matrix, so doubling a 2\times2 multiplies the determinant by 4, not 2. Also |AB|=|A||B| and |A^T|=|A|. Swapping two rows changes the sign, and two equal rows give zero (Fig 6.2).
5. Cofactor signs follow a checkerboard. The cofactor is C_{ij}=(-1)^{i+j}M_{ij}, so the signs start +,-,+ along the first row. The minor M_{ij} has no sign of its own. When expanding, pick the row or column with the most zeros, as the zero in Fig 6.2 saved a product.
6. The adjoint is the transpose of the cofactor matrix. Many students forget the transpose. For a 2\times2 matrix the shortcut is to swap the diagonal entries and negate the other two (Fig 6.3). The key identity is A\,(\operatorname{adj}A)=|A|\,I, and |\operatorname{adj}A|=|A|^{n-1}.
7. An inverse exists only when |A|\neq0. Then A^{-1}=\frac{\operatorname{adj}A}{|A|}, and |A^{-1}|=\frac{1}{|A|}. A matrix with determinant zero is singular and has no inverse, as when two rows are proportional. In Fig 6.2 a zero determinant would collapse the square onto a line.
8. Solve AX=B with X=A^{-1}B in that order. Writing BA^{-1} is wrong because the product is not commutative. If |A|=0 the system has no unique solution: it has none or infinitely many. Cramer's rule gives each unknown as a ratio of determinants, x=\frac{D_x}{D}.

6.4.1 Worked checks

Q. If A=\begin{pmatrix}1&2&3\\4&5&6\end{pmatrix} and B=\begin{pmatrix}2&1\\0&3\\1&2\end{pmatrix}, find AB.

Answer. AB is 2\times2. Row 1 by column 1: 1\cdot2+2\cdot0+3\cdot1=5. Row 1 by column 2: 1+6+6=13. Row 2 by column 1: 8+0+6=14. Row 2 by column 2: 4+15+12=31. So AB=\begin{pmatrix}5&13\\14&31\end{pmatrix}.

Q. Evaluate \begin{vmatrix}2&0&1\\1&3&2\\1&1&4\end{vmatrix} and say whether the matrix is invertible.

Answer. Expand along row 1: 2(12-2)-0+1(1-3)=20-2=18. Since 18\neq0 the matrix is non-singular, so the inverse exists. Sarrus' rule gives the same value, 24+0+1-3-4-0=18.

Q. Solve 2x+y=7 and x-y=2 using the inverse.

Answer. A=\begin{pmatrix}2&1\\1&-1\end{pmatrix}, |A|=-3, A^{-1}=\frac{1}{-3}\begin{pmatrix}-1&-1\\-1&2\end{pmatrix}.

Then X=A^{-1}\begin{pmatrix}7\\2\end{pmatrix}=\frac{1}{-3}\begin{pmatrix}-9\\-3\end{pmatrix}=\begin{pmatrix}3\\1\end{pmatrix}. So x=3,y=1, which agrees with Fig 6.3.

Practice questions

Set A — Recall (20 questions)

1. A matrix of order 5 × 3 has how many elements in a single column?a5b3c4d15
2. The determinant of [[2, 3], [1, 4]] isa11b7c5d1
3. The product of a 2 × 3 matrix and a 3 × 4 matrix has ordera3 × 3b2 × 4c4 × 2dnot defined
4. A square matrix whose determinant is zero is calledaidentitybsymmetriccorthogonaldsingular
5. For a square matrix A of order n, AI equalsaAb0cIdA^2
6. The transpose of a 2 × 3 matrix has ordera2 × 3b3 × 2c2 × 2d3 × 3
7. (AB)^T equalsaA^TB^TbABcB^TA^TdBA
8. The inverse of a square matrix A exists if and only ifaA is symmetricb|A|=0cA is nulld|A|\ne0
9. If A= [[1, 2], [3, 4]] and B= [[0, 1], [1, 0]], the element in row 2, column 1 of A+B isa3b4c5d1
10. The determinant of the 3 × 3 identity matrix isa1b0c3d9
11. A square matrix is symmetric ifaA=-A^TbA=A^{-1}c|A|=1dA=A^T
12. If A is 3 × 3 and |A|=2, then |2A| isa4b8c16d6
13. The determinant of a matrix with two identical rows isa0b1c-1d2
14. The inverse of the 2 × 2 matrix 2I isa2Ib\frac{1}{2}IcId-2I
15. A matrix all of whose elements are zero is called aaidentity matrixbscalar matrixcdiagonal matrixdnull matrix
16. The number of elements in a 3 × 4 matrix isa7b10c12d34
17. The determinant of [[1, 2], [2, 4]] isa4b0c8d-4
18. In the adjoint of [[a, b], [c, d]], the element in row 1, column 2 isa-bbbc-cdc
19. The diagonal elements of a skew-symmetric matrix area1b-1call equaldzero
20. |A^{-1}| equalsa|A|b-|A|c\frac{1}{|A|}d|A|^2

Set B — Understanding (12 questions)

1. If the determinant of [[x, 2], [3, 4]] is -2, then x isa3b2c-1d1
2. For A= [[1, 2], [3, 4]], the element in row 1, column 1 of A^2 isa1b7c10d15
3. For A= [[1, 2], [0, 1]] and B= [[1, 0], [3, 1]], the element in row 2, column 1 of AB isa3b7c1d0
4. In the inverse of [[3, 1], [5, 2]], the element in row 1, column 2 isa1b5c-1d3
5. The matrix [[k, 2], [3, 6]] is singular when k equalsa0b2c3d1
6. If A is 3 × 3 and |A|=5, then |\text{adj}A| isa25b5c125d15
7. If A is 3 × 2 and B is 2 × 3, the order of AB isa2 × 2b3 × 2c3 × 3d2 × 3
8. For the system x+y=5, x-y=1, the value of x isa2b3c4d1
9. The determinant of [[2, 5, 1], [0, 3, 7], [0, 0, 4]] isa24b14c9d0
10. If A is 2 × 2 with |A|=3, then |3A| isa9b81c6d27
11. The cofactor of the element in row 1, column 2 of [[1, 2], [3, 4]] isa3b-3c4d-4
12. If [[a, 0], [0, a]] squared equals I, a possible value of a isa0b2c-1d3

Set C — Application (8 questions)

1. Solving 2x+3y=8 and x+y=3 by Cramer's rule gives x+2y equal toa4b5c6d3
2. If A is a 3 × 3 matrix with |A|=-2, then |A^{-1}| isa2b-2c\frac{1}{2}d-\frac{1}{2}
3. If A= [[1, 1], [0, 1]], the element in row 1, column 2 of A^3 isa3b1c2d6
4. The determinant of [[1, 2, 3], [4, 5, 6], [7, 8, 9]] isa1b45c0d-6
5. For A= [[2, 1], [1, 1]], the element in row 1, column 1 of A^{-1} isa2b-1c\frac{1}{2}d1
6. The product of [[1, 2], [3, 4]] and the column [[1], [1]] is a column whose entries add up toa7b10c3d14
7. If the determinant of [[x, 2], [3, x]] is 3, a possible value of x isa9b6c3d0
8. If A is the 2 × 2 matrix with a_{ij}=i+j, then |A| isa-1b1c0d17