Matrix

A class teacher records the marks of three students in four subjects in a table of 3 rows and 4 columns. A shopkeeper with two shops records the pens, copies and bags sold in each shop in a table of 2 rows and 3 columns. When we keep only the numbers of such a table and put them in brackets, we get a matrix (plural: matrices). A matrix lets us treat a whole table of numbers as one object: we can compare two tables, add them, subtract them, or scale a whole table up or down in one step. Matrices are used in computer graphics, in economics and in solving equations, as you will see in grade 9 and 10. This chapter introduces the language of matrices, the main types of matrix, and the three operations of addition, subtraction and multiplication by a number.

5.1 Matrix: elements, rows, columns and order

Here are the marks of three students of a school in Hetauda.

Student	Maths	Science	English	Opt. Maths
Anish	78	65	70	82
Bipana	85	90	72	88
Chandra	60	74	81	69

Keeping only the numbers, in the same places, gives the matrix M drawn below.

Matrices are named with capital letters A, B, M, \ldots and their elements with the matching small letter and two little numbers. The element in the ith row and jth column of A is written a_{ij}: row first, column second. In the marks matrix, m_{23}=72 and m_{34}=69. A matrix of order 2\times3 looks, in general, like this:

	column 1	column 2	column 3
row 1	a_{11}	a_{12}	a_{13}
row 2	a_{21}	a_{22}	a_{23}

We write it briefly as A=[a_{ij}]_{2\times3}.

Matrices are everywhere once you look for them. A black-and-white photograph on a phone is stored as a matrix: the screen is divided into tiny squares called pixels, and each element of the matrix is the brightness of one pixel, from 0 (black) to 255 (white). A photo 1200 pixels wide and 800 pixels high is a matrix of order 800\times1200 with 960,000 elements. Brightening the photo adds the same number to every element; making a "negative" replaces each element a by 255-a. The operations of this chapter are exactly what photo-editing apps do.

Writing a matrix inside a line of text. A matrix needs a rectangle of space, and a line of print has none. So in running text this book sometimes lists the rows in order, each row in round brackets: [(3,5,7),(4,6,8)] means the 2\times3 matrix whose first row is 3, 5, 7 and whose second row is 4, 6, 8. In your own work, always draw the full rectangular array, as in the figures.

5.2 Types of matrices: row, column, zero, rectangular, square, diagonal, scalar, unit, equal, symmetric and triangular

Matrices are named by their shape and by the pattern of their elements. The figure shows one example of each type; the table below it gives the definitions.

Type	Definition	Example
Row matrix	only one row; order 1\times n	[(2,5,-1)]
Column matrix	only one column; order m\times1	[(3),(0),(7)]
Zero (null) matrix O	every element is 0; any order	[(0,0),(0,0)]
Rectangular matrix	number of rows \neq number of columns	order 2\times3
Square matrix	number of rows = number of columns; order n\times n, or "order n"	order 2\times2
Diagonal matrix	a square matrix whose elements off the leading diagonal are all 0	[(2,0,0),(0,-1,0),(0,0,5)]
Scalar matrix	a diagonal matrix whose diagonal elements are all equal	[(3,0),(0,3)]
Unit (identity) matrix I	a scalar matrix whose diagonal elements are all 1	[(1,0),(0,1)]
Symmetric matrix	a square matrix with a_{ij}=a_{ji} for every i and j	[(1,4,5),(4,2,6),(5,6,3)]
Upper triangular matrix	a square matrix whose elements below the leading diagonal are all 0	[(1,2,3),(0,4,5),(0,0,6)]
Lower triangular matrix	a square matrix whose elements above the leading diagonal are all 0	[(1,0,0),(2,3,0),(4,5,6)]

The leading (principal) diagonal of a square matrix runs from the top-left corner to the bottom-right corner: the elements a_{11}, a_{22}, a_{33}, \ldots. Only square matrices have one, which is why the last six types in the table are all square.

Symmetric matrices and the transpose

In a symmetric matrix the leading diagonal acts like a mirror: the element in row 1, column 2 equals the element in row 2, column 1, and so on. Interchanging the rows and columns of a matrix A (row 1 becomes column 1, row 2 becomes column 2, …) gives its transpose, written A^T or A'. A matrix is symmetric exactly when it equals its own transpose, A^T=A.

A family of special square matrices

Some types sit inside others:

• every unit matrix is a scalar matrix (all its diagonal elements are equal to 1);
• every scalar matrix is a diagonal matrix;
• every diagonal matrix is both upper and lower triangular, and it is also symmetric.

So I deserves all five names. But a symmetric matrix need not be diagonal, and a triangular matrix need not be symmetric.

Equal matrices

Matrices of different orders are never equal, even if they contain the same numbers: the row matrix [(1,2)] is not equal to the column matrix [(1),(2)].

5.3 Addition and subtraction of matrices and the properties of matrix addition

Addition and subtraction

Two matrices can be added or subtracted only if they have the same order. Then we add (or subtract) the elements that are in the same position.

Properties of matrix addition

Let A, B and C be matrices of the same order, and O the zero matrix of that order.

Property	Statement
Closure	A+B is again a matrix of the same order
Commutative	A+B=B+A
Associative	(A+B)+C=A+(B+C)
Additive identity	A+O=O+A=A; the zero matrix changes nothing
Additive inverse	A+(-A)=O, where -A has every element of A with its sign changed

These hold because each element of a sum is an ordinary sum of numbers, and numbers obey the same laws. Matrix subtraction is not commutative: in general A-B\neq B-A.

The additive inverse also explains subtraction: A-B=A+(-B). To subtract B, change the sign of every element of B and add. This is the same rule you used for subtracting polynomials, and it is the safest way to avoid sign errors when an element of B is negative, as with 3-(-3)=3+3=6 in the figure above.

5.4 Multiplication of a matrix by a scalar

In matrix work an ordinary number is called a scalar. If every price in a list goes up by 10%, every element is multiplied by the same number 1.1. This is scalar multiplication.

Properties of scalar multiplication

For matrices A, B of the same order and scalars k, l:

Property	Statement
Distributive over matrix addition	k(A+B)=kA+kB
Distributive over scalar addition	(k+l)A=kA+lA
Associative	k(lA)=(kl)A
Multiplying by 1 and by 0	1A=A and 0A=O

The matrices used in the practice questions

Some practice questions below refer to the matrices in this figure.

Chapter summary

• A matrix is a rectangular array of numbers in rows and columns; a matrix with m rows and n columns has order m\times n and mn elements; a_{ij} is the element in row i, column j.
• Row 1\times n, column m\times1, zero (all 0), rectangular (m\neq n), square (m=n).
• In a square matrix: diagonal (zero off the leading diagonal), scalar (diagonal with equal diagonal elements), unit I (scalar with 1s), symmetric (a_{ij}=a_{ji}), upper/lower triangular (zeros below/above the leading diagonal).
• Equal matrices have the same order and equal corresponding elements.
• A\pm B=[a_{ij}\pm b_{ij}], possible only for the same order.
• Addition is closed, commutative and associative, with identity O and inverse -A; subtraction is not commutative.
• kA=[k\,a_{ij}]; k(A+B)=kA+kB, (k+l)A=kA+lA, k(lA)=(kl)A, 1A=A, 0A=O.

Practice questions

The matrices P, Q, R, S, T, U, V and W are those in the figure above.

Group A — Objective (1 mark each)

1. A matrix with 3 rows and 2 columns has order [1]a2\times3b3\times2c6d3+2
2. The number of elements in a matrix of order 4\times3 is [1]a7b12c43d1
3. In the matrix R, the element r_{23} is [1]a2b4c-1d3
4. The matrix V is a [1]acolumn matrixbrow matrixcsquare matrixdzero matrix
5. The best name for the matrix T is [1]aunit matrixbscalar matrixcrow matrixdzero matrix
6. The matrix S is [1]asymmetricba unit matrixca row matrixdlower triangular
7. The matrix U is [1]alower triangularbupper triangularcdiagonaldscalar
8. If [(x,2),(3,4)]=[(5,2),(3,4)], then x is [1]a2b3c4d5
9. P+Q equals [1]a[(5,3),(-2,6)]b[(-1,-5),(2,4)]c[(6,-4),(0,5)]d[(5,4),(-2,6)]
10. 2P equals [1]a[(4,-2),(0,10)]b[(4,-1),(0,5)]c[(2,-1),(0,5)]d[(4,1),(0,10)]
11. Which pair of matrices can be added? [1]aP and RbP and QcV and WdR and S
12. The additive inverse of P is [1]a[(-2,1),(0,-5)]b[(2,1),(0,5)]c[(-2,-1),(0,-5)]d[(\frac{1}{2},-1),(0,\frac{1}{5})]

Answers to Group A

1. (b) — rows first, then columns.

2. (b) — 4\times3=12 elements.

3. (b) — row 2, column 3 of R is 4.

4. (b) — V has one row and three columns.

5. (b) — a diagonal matrix with every diagonal element equal to 5 (not 1, so not the unit matrix).

6. (a) — s_{12}=s_{21}=2.

7. (b) — all the elements below the leading diagonal are 0.

8. (d) — corresponding elements: x=5.

9. (a) — [(2+3,\ -1+4),(0-2,\ 5+1)].

10. (a) — every element is doubled.

11. (b) — only P and Q have the same order, 2\times2.

12. (a) — change the sign of every element: -P.

Group B — Short answer (2 marks each)

1. Write the order of R and its elements r_{12} and r_{21}. [2]
2. Construct the 2\times2 matrix A=[a_{ij}] with a_{ij}=2i+j. [2]
3. Find x and y if [(x+3,\ 4),(6,\ 2y)]=[(8,\ 4),(6,\ 10)]. [2]
4. Find P-Q and Q-P. [2]
5. Find 3Q-2P. [2]
6. If [(a,\ 3),(b,\ 5)] is symmetric and a+b=7, find a and b. [2]
7. Find the matrix X such that X+P=Q. [2]
8. Write the unit matrix of order 3 and the scalar matrix of order 2 whose diagonal elements are -3. [2]
9. Verify that P+Q=Q+P. [2]
10. If A=[(4,-6),(2,8)], find \frac{1}{2}A and -A. [2]

Answers to Group B

1. R has 2 rows and 3 columns, so its order is 2\times3; r_{12}=0 and r_{21}=3.

2. a_{11}=3, a_{12}=4, a_{21}=5, a_{22}=6, so A=[(3,4),(5,6)].

3. x+3=8 gives x=5; 2y=10 gives y=5.

4. P-Q=[(2-3,\ -1-4),(0+2,\ 5-1)]=[(-1,-5),(2,4)] and Q-P=[(1,5),(-2,-4)]=-(P-Q).

5. 3Q=[(9,12),(-6,3)] and 2P=[(4,-2),(0,10)], so 3Q-2P=[(5,14),(-6,-7)].

6. Symmetric means a_{12}=a_{21}, so b=3; then a=7-3=4.

7. X=Q-P=[(1,5),(-2,-4)]. Check: X+P=[(3,4),(-2,1)]=Q ✓.

8. I=[(1,0,0),(0,1,0),(0,0,1)] and [(-3,0),(0,-3)].

9. P+Q=[(5,3),(-2,6)] and Q+P=[(3+2,\ 4-1),(-2+0,\ 1+5)]=[(5,3),(-2,6)]; they are equal.

10. \frac{1}{2}A=[(2,-3),(1,4)] and -A=[(-4,6),(-2,-8)].

Group C — Long answer (4 marks each)

1. If A=[(1,2),(3,4)], B=[(5,-1),(0,2)] and C=[(-2,3),(4,1)], verify that (A+B)+C=A+(B+C). [4]
2. Find x, y and z if [(x+y,\ 4),(z,\ x-y)]+[(1,\ -2),(3,\ 2)]=[(9,\ 2),(5,\ 4)]. [4]
3. Two shops sell pens, copies and bags. In Baisakh their sales were B=[(120,80,15),(95,60,20)] and in Jestha J=[(140,70,18),(110,75,12)] (rows: shops; columns: pens, copies, bags). (i) Find the total sales of the two months. (ii) Find the change from Baisakh to Jestha and explain the negative entries. (iii) If the target for Asar is double the Jestha sales, find the target matrix. [4]
4. If A=[(2,0),(-1,3)] and B=[(2,4),(3,-1)], verify that 3(A+B)=3A+3B and find the matrix X such that 2X-A=B. [4]
5. Classify each of the matrices P, Q, R, S, T, U, V, W by every type that applies to it. Then find T-5I, where I is the unit matrix of order 3, and comment on the result. [4]

Answers to Group C

1. A+B=[(6,1),(3,6)], so (A+B)+C=[(4,4),(7,7)]. B+C=[(3,2),(4,3)], so A+(B+C)=[(1+3,\ 2+2),(3+4,\ 4+3)]=[(4,4),(7,7)]. The two sides are equal, so addition is associative here.

2. The left side is [(x+y+1,\ 2),(z+3,\ x-y+2)]. Equating: x+y+1=9, so x+y=8; x-y+2=4, so x-y=2; z+3=5, so z=2. Adding the first two, 2x=10, so x=5 and y=3. The element 2=2 agrees.

3. (i) B+J=[(260,150,33),(205,135,32)]. (ii) J-B=[(20,-10,3),(15,15,-8)]: the first shop sold 10 fewer copies and the second shop 8 fewer bags in Jestha; every other entry rose. (iii) 2J=[(280,140,36),(220,150,24)].

4. A+B=[(4,4),(2,2)], so 3(A+B)=[(12,12),(6,6)]. 3A=[(6,0),(-3,9)] and 3B=[(6,12),(9,-3)], so 3A+3B=[(12,12),(6,6)]; verified. From 2X-A=B: 2X=A+B=[(4,4),(2,2)], so X=[(2,2),(1,1)]. Check: 2X-A=[(4-2,\ 4-0),(2+1,\ 2-3)]=[(2,4),(3,-1)]=B ✓.

5. P: square, upper triangular (its element below the diagonal is 0). Q: square. R: rectangular, 2\times3. S: square, symmetric. T: square, diagonal, scalar, symmetric, upper and lower triangular. U: square, upper triangular. V: row, rectangular. W: column, rectangular. T-5I=[(5,0,0),(0,5,0),(0,0,5)]-[(5,0,0),(0,5,0),(0,0,5)]=O, the zero matrix; so T=5I: every scalar matrix is a scalar multiple of the unit matrix.