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Analytical Chemistry Technology professionals in Kenya often encounter complex data sets and systems of linear equations in quantitative analysis, calibration, and instrument calibration. Mastery of matrices enables efficient handling of such data, simplifying calculations and enhancing accuracy in problem solving. This chapter introduces matrices, their types, orders, and operations, emphasizing practical applications relevant to analytical chemistry contexts.
A matrix is a rectangular array of numbers arranged in rows and columns and enclosed in brackets. Each number is an element of the matrix, and the element in row i and column j is written a(i,j). Laboratory data arrive in this form naturally: absorbance readings of several samples at several wavelengths, reagent volumes for several standard solutions, or calibration coefficients of a set of equations. Writing them as a matrix allows the whole table to be added, scaled or multiplied in one step.
Example: absorbance readings of three samples (rows) at two wavelengths (columns) form a matrix with 3 rows and 2 columns.
[ 0.12 0.20 ]
[ 0.35 0.41 ]
[ 0.48 0.52 ]
The element in row 2 and column 1 is 0.35.
Matrices are classified by properties that decide what can be done with them. Singular and non-singular describe whether a square matrix has an inverse, an identity matrix is the matrix that leaves others unchanged when multiplied, and an echelon matrix is one arranged in a staircase pattern ready for solving equations. The names row, column and square matrix describe shape and are given under Order of matrices, and the zero matrix is given under Addition and subtraction.
A singular matrix is a square matrix with a determinant of zero. It has no inverse. Its rows (or columns) are not independent: one row is a multiple of another, or a sum of others, so it carries no new information. A set of calibration equations whose coefficient matrix is singular has no single answer.
The determinant of a 2 × 2 matrix with elements a, b in the first row and c, d in the second row is a × d − b × c. Its calculation and use are taught in Determinant and inverse of 2x2 matrix.
Example: find whether the matrix below is singular.
[ 2 4 ]
[ 1 2 ]
Determinant = (2 × 2) − (4 × 1) = 4 − 4 = 0. The matrix is singular, because the first row is exactly twice the second row.
A non-singular matrix is a square matrix whose determinant is not zero. It has an inverse, so a set of equations with this coefficient matrix has exactly one solution. This is the case the analyst wants for a calibration with two unknowns.
Example: the matrix of calibration coefficients
[ 1.2 0.5 ]
[ 0.4 1.1 ]
has determinant (1.2 × 1.1) − (0.5 × 0.4) = 1.32 − 0.20 = 1.12. Since 1.12 is not zero, the matrix is non-singular and has an inverse.
An identity matrix is a square matrix with 1 on the main diagonal (from top left to bottom right) and 0 everywhere else. It is written I, or I₂, I₃ to show its order. It does for matrices what the number 1 does for numbers: multiplying any matrix by an identity matrix of the right order leaves it unchanged, so A × I = I × A = A.
I₂ = [ 1 0 ] I₃ = [ 1 0 0 ]
[ 0 1 ] [ 0 1 0 ]
[ 0 0 1 ]
Example: multiply the calibration matrix by I₂ using the row-by-column rule shown under Matrix multiplication.
[ 1.2 0.5 ] [ 1 0 ] [ 1.2 0.5 ]
[ 0.4 1.1 ] × [ 0 1 ] = [ 0.4 1.1 ]
First row: (1.2 × 1) + (0.5 × 0) = 1.2 and (1.2 × 0) + (0.5 × 1) = 0.5. Second row: (0.4 × 1) + (1.1 × 0) = 0.4 and (0.4 × 0) + (1.1 × 1) = 1.1. The product equals the original matrix.
A matrix is in echelon form (row echelon form) when it has a staircase shape:
Echelon form Not echelon form
[ 2 1 3 ] [ 0 1 2 ]
[ 0 4 5 ] [ 1 0 3 ]
[ 0 0 6 ]
The second matrix fails because the leading element of the second row is to the left of that of the first row. In reduced echelon form every leading element is also 1 and has zeros above it as well as below it.
Any matrix can be brought to echelon form by row operations that do not change the solutions of the equations it stands for: swap two rows, multiply a row by a non-zero number, or add a multiple of one row to another.
Example: reduce the matrix below to echelon form.
[ 1 2 3 ]
[ 2 5 7 ]
[ 3 7 10 ]
Step 1: replace row 2 by row 2 − 2 × row 1: (2 − 2, 5 − 4, 7 − 6) = (0, 1, 1). Replace row 3 by row 3 − 3 × row 1: (3 − 3, 7 − 6, 10 − 9) = (0, 1, 1).
[ 1 2 3 ]
[ 0 1 1 ]
[ 0 1 1 ]
Step 2: replace row 3 by row 3 − row 2: (0, 0, 0).
[ 1 2 3 ]
[ 0 1 1 ]
[ 0 0 0 ]
The result is in echelon form. It has a row of zeros, which shows that the original matrix is singular (its determinant is 0). Echelon form is also the basis of Gaussian elimination, another way of solving simultaneous equations.
The order of a matrix is its size, written as the number of rows followed by the number of columns: a matrix with m rows and n columns has order m × n and contains m × n elements. Rows are always named first, so a 2 × 3 matrix and a 3 × 2 matrix are different. The order decides whether two matrices can be added or multiplied.
[0.12 0.35 0.48] form a 1 × 3 matrix.Row matrix 1 × 3 Column matrix 4 × 1
[ 0.12 0.35 0.48 ] [ 0.10 ]
[ 0.25 ]
[ 0.50 ]
[ 0.75 ]
Example: find the order of the matrix below.
[ 2 5 7 ]
[ 1 3 0 ]
It has 2 rows and 3 columns, so its order is 2 × 3, and it has 2 × 3 = 6 elements.
The operations on matrices are addition, subtraction, multiplication by a scalar (an ordinary number) and multiplication of two matrices. Each is defined only when the orders of the matrices allow it. Two matrices are equal only when they have the same order and every pair of matching elements is equal.
Matrices of the same order are added or subtracted by adding or subtracting the elements in matching positions: each element of the answer is a(i,j) + b(i,j), or a(i,j) − b(i,j). The answer has the same order as the two matrices. Addition gives the same answer in either order (A + B = B + A), but subtraction does not, since A − B and B − A have opposite signs.
A zero matrix, written O, has every element 0 and may have any order. Adding it changes nothing (A + O = A), and subtracting a matrix from itself gives it (A − A = O).
Example: add the matrices.
[ 1 2 ] [ 5 6 ] [ 6 8 ]
[ 3 4 ] + [ 7 8 ] = [ 10 12 ]
Example: subtract the matrices.
[ 10 15 20 ] [ 1 2 3 ] [ 9 13 17 ]
[ 25 30 35 ] − [ 4 5 6 ] = [ 21 25 29 ]
A 2 × 2 matrix cannot be added to a 2 × 3 matrix, because the positions do not match.
To multiply a matrix by a scalar k, multiply every element by k. The order does not change. It is used to convert units, to apply a dilution factor, and with addition to find means.
Example: multiply by k = 3.
[ 2 4 ] [ 6 12 ]
3 × [ 6 8 ] = [ 18 24 ]
Example: readings of 0.1, 0.2 and 0.3 mol/L were made on samples diluted 10 times; multiply the row by 10 to give the original concentrations.
10 × [ 0.1 0.2 0.3 ] = [ 1 2 3 ]
Example: the mean of two replicate sets of absorbance readings, A and B, is ½ × (A + B).
A = [ 0.12 0.35 ] B = [ 0.14 0.33 ]
[ 0.48 0.60 ] [ 0.50 0.58 ]
A + B = [ 0.26 0.68 ] ½ × (A + B) = [ 0.13 0.34 ]
[ 0.98 1.18 ] [ 0.49 0.59 ]
A scalar also distributes over addition: k × (A + B) = k × A + k × B.
Two matrices are compatible for an operation when their orders allow it.
A quick check is to write the two orders side by side. For (2 × 3) × (3 × 2) the inner numbers are both 3, so the product exists, and the outer numbers give its order, 2 × 2.
| Order of A | Order of B | A + B | A × B | B × A |
|---|---|---|---|---|
| 2 × 3 | 2 × 3 | Possible, 2 × 3 | Not possible (3 columns, 2 rows) | Not possible (3 columns, 2 rows) |
| 2 × 3 | 3 × 2 | Not possible | Possible, 2 × 2 | Possible, 3 × 3 |
| 1 × 2 | 3 × 1 | Not possible | Not possible (2 columns, 3 rows) | Possible, 3 × 2 |
| 2 × 3 | 2 × 2 | Not possible | Not possible (3 columns, 2 rows) | Possible, 2 × 3 |
| 3 × 3 | 3 × 3 | Possible, 3 × 3 | Possible, 3 × 3 | Possible, 3 × 3 |
The table shows that a product may exist in one order and not in the other. Even when both A × B and B × A exist, they may differ in order, and for matrices of the same order they are usually different, as the next section shows.
To multiply A (order m × n) by B (order n × p), first check compatibility. The product C = A × B has order m × p, and its element in row i and column j is found by multiplying the elements of row i of A by the matching elements of column j of B and adding the products:
c(i,j) = a(i,1) × b(1,j) + a(i,2) × b(2,j) + ... + a(i,n) × b(n,j)
Example: a 1 × 2 matrix times a 2 × 1 matrix gives a 1 × 1 matrix. The number of columns of A (2) equals the number of rows of B (2).
[ 1 2 ] × [ 3 ] = [ (1 × 3) + (2 × 4) ] = [ 11 ]
[ 4 ]
Example: multiply two 2 × 2 matrices.
[ 1 3 ] [ 5 7 ] [ 23 31 ]
[ 2 4 ] × [ 6 8 ] = [ 34 46 ]
Reversing the order gives a different answer, so matrix multiplication is not commutative:
[ 5 7 ] [ 1 3 ] [ 19 43 ]
[ 6 8 ] × [ 2 4 ] = [ 22 50 ]
Example: multiply a 2 × 3 matrix by a 3 × 2 matrix. The product is 2 × 2.
[ 2 0 1 ] [ 1 2 ] [ 6 4 ]
[ 3 1 4 ] × [ 0 1 ] = [ 19 7 ]
[ 4 0 ]
Example: a laboratory prepares two standard solutions from three reagents. The matrix V gives the volumes in millilitres (rows are the solutions, columns the reagents), and the column matrix P gives assumed prices of the reagents in Ksh per millilitre. V × P is a 2 × 3 matrix times a 3 × 1 matrix, so the result is 2 × 1: the cost in Ksh of each solution.
[ 10 5 20 ] [ 2.5 ] [ 69.0 ]
[ 15 0 10 ] × [ 4.0 ] = [ 49.5 ]
[ 1.2 ]
So solution 1 costs Ksh 69.0 and solution 2 costs Ksh 49.5.
$$ \begin{bmatrix} 4 & 7 & 1 \\ 2 & 5 & 9 \\ 3 & 6 & 8 \\ 1 & 0 & 4 \end{bmatrix} $$
(3 marks)
$$ A = \begin{bmatrix} 2 & 4 \\ 6 & 8 \end{bmatrix}, \quad B = \begin{bmatrix} 1 & 3 \\ 5 & 7 \end{bmatrix} $$
(4 marks)
$$ A = \begin{bmatrix} 1 & 0 & 2 \\ -1 & 3 & 1 \end{bmatrix} $$
by scalar \( k = 5 \).
(4 marks)
$$ A = \begin{bmatrix} 3 & 5 \\ 2 & 4 \end{bmatrix}, \quad B = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} $$
(6 marks)
$$ A = \begin{bmatrix} 1 & 2 & 3 \end{bmatrix}, \quad B = \begin{bmatrix} 4 \\ 5 \\ 6 \end{bmatrix} $$
(5 marks)
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Create a free accountThis chapter introduced matrices by defining their types, including singular, non-singular, identity, and echelon matrices, and explaining their distinct properties. The concept of matrix order was discussed to clarify the dimensions that define matrices. Basic matrix operations such as addition, subtraction, multiplication by a scalar, and the conditions for compatibility in matrix multiplication were explored in detail. The chapter then focused on the determinant and inverse of a 2x2 matrix, providing the foundation for solving matrix equations. Using these concepts, the method of solving simultaneous equations with two unknowns through matrices was demonstrated. Finally, the chapter covered eigenvalues and eigenvectors, highlighting their significance in matrix theory and applications. Together, these topics form a comprehensive understanding of matrix algebra essential for scientific and engineering problem solving.
Calculate the sum of matrices $$A = \begin{bmatrix} 2 & 3 \\ 5 & 7 \end{bmatrix}$$ and $$B = \begin{bmatrix} 4 & 1 \\ 6 & 2 \end{bmatrix}.$$ (2 marks)
Multiply the matrix $$C = \begin{bmatrix} 3 & 0 \\ 1 & 4 \end{bmatrix}$$ by the scalar 5.
(2 marks)
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