Matrix Formulas

 

A matrix is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns. The numbers are called the elements, or entries, of the matrix. The elements of an array can be added, subtracted and multiplied, and can be used to represent linear transformations and vectors and solve equations.

 

Matrix Dimensions

 

Terms
Dimensions of a matrixThe number of rows and columns of the matrix.
ElementAn individual item in a matrix.

 

The dimensions of a matrix are the size of the matrix: the number of rows and the number of columns of the matrix (in this order).

If a matrix has m rows and n columns, the matrix is said to be a 'm by n' matrix, written 'm × n'.

Consider matrix A:

    \[A = \begin{bmatrix} 1 & 5 & -7 \\ 7 & 9 & 12 \end{bmatrix} \quad \]

Matrix A is a 2 x 3 matrix with 2 rows and 3 columns.

A matrix with n rows and n columns is called a square matrix of order n.

Consider matrix B:

    \[B = \begin{bmatrix} 3 & 6 \\ 5 & 7 \end{bmatrix} \quad \]

Matrix B is a square matrix of order 2.

An ordinary number can be regarded as a 1 × 1 matrix, thus 7 can be thought of as the matrix

    \[\begin{bmatrix} 7 \end{bmatrix} \quad \]

.

 

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Matrix Elements

In common notation, a capital letter denotes a matrix, and the corresponding small letter with a double subscript describes an element of the matrix.

A matrix element is simply a matrix entry. Each element in a matrix is identified by naming the row and column in which it appears. The element in row i and column j of Matrix A is denoted as ai,j.

Consider matrix C:

 

    \[C = \begin{bmatrix} -5 & 10 \\ 12 & 6 \\ 5 & 7 \end{bmatrix} \quad \]

 

If C is the 2 × 3 matrix shown above, then a1,1 = -5, a1,2 = 10, a2,1 = 12, a2,2 = 6, a3,1 = 5, and a3,2 = 7. Thus, a1,1 is the entry in the first row and the first column, a2,1 is the entry in the second row and first column, etc.

 

Types of Matrices

 

Name Size Description Example
Row Matrix 1 x n A matrix is said to be a row matrix if it has only one row. It is sometimes used to represent a vector. 1 x 4 row matrix

    \[ \begin{bmatrix} -5 & 10 & 6 & 8 \end{bmatrix} \quad \]

Column Matrix n x 1 A matrix is said to be a column matrix if it has only one column. It is sometimes used to represent a vector. 3 x 1 column matrix

    \[ \begin{bmatrix} -5\\ 12\\ 5 \end{bmatrix} \quad \]

Rectangular Matrix n x m A matrix is said to be rectangular if the number of rows is not equal to the number of columns. 2 x 3 rectangular matrix

    \[ \begin{bmatrix} -5 & 7 & 12\\ 2 & 3 & 5\\ \end{bmatrix} \quad \]

Square Matrix n x n A matrix with the same number of rows and columns, sometimes used to represent a linear transformation from a vector space to itself, such as reflection, rotation or shearing. 3 x 3 square matrix

    \[ \begin{bmatrix} -5 & 10 & 6\\ -5 & 7 & 12\\ 2 & 3 & 5\\ \end{bmatrix} \quad \]

Diagonal Matrix n x n A square matrix is said to be diagonal if at least one element of principal diagonal is non-zero and all the other elements are zero. 3 x 3 diagonal matrix

    \[ \begin{bmatrix} -5 & 0 & 0\\ 0 & -7 & 0\\ 0 & 0 & 2\\ \end{bmatrix} \quad \]

Scalar Matrix n x n A diagonal matrix is said to be scalar if all of its diagonal elements are the same. 3 x 3 scalar matrix

    \[ \begin{bmatrix} 7 & 0 & 0\\ 0 & 7 & 0\\ 0 & 0 & 7\\ \end{bmatrix} \quad \]

Identity/Unit Matrix n x n A diagonal matrix is said to be identity if all of its diagonal elements are equal to one, denoted by I. 2 x 2 identity matrix

    \[I = \begin{bmatrix} 1 & 0\\ 0 & 1 \end{bmatrix} \quad \]

Lower Triangular Matrix n x n A square matrix is said to be triangular if all of its elements above the principal diagonal are zero (lower triangular matrix). 3 x 3 lower triangular matrix

    \[ \begin{bmatrix} 7 & 0 & 0\\ 4 & 8 & 0\\ 9 & 6 & 7\\ \end{bmatrix} \quad \]

Upper Triangular Matrix n x n A square matrix is said to be triangular if all of its elements below the principal diagonal are zero (upper triangular matrix). 3 x 3 upper triangular matrix

    \[ \begin{bmatrix} 7 & 0 & 0\\ 4 & 8 & 0\\ 9 & 6 & 7\\ \end{bmatrix} \quad \]

Null/Zero Matrix n x n A matrix is said to be a null or zero matrix if all of its elements are equal to zero. It is denoted by O. 2 x 2 zero matrix

    \[O = \begin{bmatrix} 0 & 0\\ 0 & 0 \end{bmatrix} \quad \]

 

Matrix Row Operations

Matrices can be added, subtracted and multiplied as individual entities, giving rise to important mathematical systems known as matrix algebras. Matrices occur naturally in simultaneous equations. In the following equations for the variables x and y,

 

    \[\ 5x + 3y = 10\]

    \[\ 4x + 2y = 12\]

 

the array of numbers

    \[ \begin{bmatrix} 5 & 3 \\ 4 & 2 \end{bmatrix} \quad \]

 

is a matrix where its elements are the coefficients of the variables. The solutions to the equations depend on these numbers and their arrangement. If 4 and 2 in the second row were interchanged, the solution would not be the same.

We use matrices to list data or to represent systems. We can perform operations on matrices because their elements/entries are numbers.

 

Application
Matrices have wide applications in engineering, physics, economics, and statistics as well as in various branches of mathematics. Some of them merely take advantage of the compact representation of a set of numbers in a matrix.

Matrices have also come to have important applications in computer graphics, where they have been used to represent rotations and other transformations of images.
Matrices over a polynomial ring are important in the study of control theory. 
Chemistry makes use of matrices in various ways, particularly since the use of quantum theory to discuss molecular bonding and spectroscopy.
Matrices are used throughout the field of machine learning in the description of algorithms and processes such as the input data variable (X) when training an algorithm.

 

Adding and Subtracting Matrices

Addition and subtraction require that the matrices be the same dimensions. The resultant matrix is also of the same dimension.

When performing addition, add each element in the first matrix to the corresponding element in the second matrix.

 

    \[ \begin{bmatrix} a_{1} & a_{2} \\ a_{3} & a_{4} \end{bmatrix} + \begin{bmatrix} b_{1} & b_{2} \\ b_{3} & b_{4} \end{bmatrix} = \begin{bmatrix} a_{1} + b_{1} & a_{2} + b_{2} \\ a_{3} + b_{3} & a_{4} + b_{4} \end{bmatrix} \quad \]

 

Consider matrix A and matrix B:

 

    \[A = \begin{bmatrix} 1 & 5  \\ 7 & 9 \end{bmatrix} ,   B = \begin{bmatrix} 3 & 6 \\ 5 & 7 \end{bmatrix} \quad \]

 

    \[ A + B = \begin{bmatrix} 1 + 3 & 5 + 6 \\ 7 + 5 & 9 + 7 \end{bmatrix} = \begin{bmatrix} 4 & 11\\ 12 & 16 \end{bmatrix} \quad \]

 

Note that the element a1 in matrix A, is added up to element b1 in matrix B, to produce element a1 + b1 = 1 + 3 = 4 in the resultant matrix. Also note that both matrices being added are 2 x 2 matrices, and the resulting matrix is also 2 x 2 matrix.

When performing subtraction, subtract each element in the second matrix from the corresponding element in the first matrix.

 

    \[ \begin{bmatrix} a_{1} & a_{2} \\ a_{3} & a_{4} \end{bmatrix} - \begin{bmatrix} b_{1} & b_{2} \\ b_{3} & b_{4} \end{bmatrix} = \begin{bmatrix} a_{1} - b_{1} & a_{2} - b_{2} \\ a_{3} - b_{3} & a_{4} - b_{4} \end{bmatrix} \quad \]

 

Consider matrix A and matrix B again:

 

    \[\ A = \begin{bmatrix} 1 & 5  \\ 7 & 9 \end{bmatrix} ,  B = \begin{bmatrix} 3 & 6 \\ 5 & 7 \end{bmatrix} \quad \]

 

    \[ A + B = \begin{bmatrix} 1 - 3 & 5 - 6 \\ 7 - 5 & 9 - 7 \end{bmatrix} = \begin{bmatrix} -2 & -1\\ 2 & 2 \end{bmatrix} \quad \]

 

Be careful when subtracting with signed numbers!

 

Scalar Matrix Multiplication

When we work with matrices, we refer to real numbers as scalars.  The term scalar multiplication refers to the product of a real number and a matrix.  In scalar multiplication, each entry in the matrix is multiplied by the given scalar.

 

    \[ \lambda. \begin{bmatrix} a_{1} & a_{2} \\ a_{3} & a_{4} \end{bmatrix}  = \begin{bmatrix} \lambda.a_{1}  & \lambda.a_{2} \\ \lambda.a_{3} & \lambda.a_{4} \end{bmatrix} \quad \]

 

For example, given that

 

    \[A = \begin{bmatrix} 1 & 5 & -7 \\ 7 & 9 & 12 \end{bmatrix} \quad \]

 

let's find 5A.

 

To find 5A, simply multiply each matrix element by 5

 

    \[ A = 5. \begin{bmatrix} 1 & 5 & -7 \\ 7 & 9 & 12 \end{bmatrix} \quad \]

 

    \[ = \begin{bmatrix} 5.1 &5.5 & 5.-7 \\ 5.7 & 5.9 & 5.12 \end{bmatrix} \quad \]

 

    \[ = \begin{bmatrix} 5 &25 & -35 \\ 35 &45 & 60 \end{bmatrix}\quad \]

 

Scalar multiplication as repeated addition

Remember that to add (or subtract) two matrices, we can simply add (or subtract) the corresponding elements.

Let's consider the repeated addition of a matrix:

 

    \[A = \begin{bmatrix} 7 & 5  \\ 3 & 4 \end{bmatrix} \quad \]

 

Find

    \[ A + A + A \]

.

    \[ A + A + A  = \begin{bmatrix} 7 & 5  \\ 3 & 4 \end{bmatrix} + \begin{bmatrix} 7 & 5  \\ 3 & 4 \end{bmatrix} + \begin{bmatrix} 7 & 5  \\ 3 & 4 \end{bmatrix} \quad \]

 

    \[ = \begin{bmatrix} 7 + 7 + 7 & 5 + 5 + 5  \\ 3 + 3 + 3 & 4 + 4 + 4 \end{bmatrix} \quad \]

 

    \[ = \begin{bmatrix} 3. 7 & 3.5  \\ 3. 3 & 3.4 \end{bmatrix} \quad \]

 

    \[ = 3. \begin{bmatrix} 7 & 5  \\ 3 & 4 \end{bmatrix} \quad \]

 

    \[ = 3A \]

 

Multiplying Matrices

When multiplying matrices, the elements of the rows in the first matrix are multiplied with corresponding columns in the second matrix.

 

What is a Matrix Product?
Schematic depiction of the matrix product AB of two matrices A and B.

 

If A is a m x n matrix and B is a n × p matrix, the result AB of their multiplication is a m × p matrix defined only if the number of columns n in A is equal to the number of rows n in B.

    \[ (m \times n)\cdot(n \times p) = (m \times p) \]

 

What is the relationship between the dimensions of two matrices and the dimensions of their product?
The relationship between the dimensions of two matrices and the dimensions of their product

 

The dot product

Consider the 2 x 3 matrix A:

    \[\ A = \begin{bmatrix} a & b & c \\ d & e & f \end{bmatrix} \quad \]

 

and the 3 x 2 matrix B:

 

    \[\ B = \begin{bmatrix} g & h \\ i & j \\ k & l \end{bmatrix} \quad \]

 

The reusltant matrix

    \[ C = AB \]

will be a 2 x 2 matrix:

 

    \[\ C = \begin{bmatrix} ag + bi + ck & ah + bj + cl  \\ dg + ei + fk & dh + ej + fl \end{bmatrix} \quad \]

 

Consider matrix A and matrix B:

 

    \[\ A = \begin{bmatrix} 3 & 5  \\ 7 & -2 \end{bmatrix} ,  B = \begin{bmatrix} -4 & 6 \\ 5 & 7 \end{bmatrix} \quad \]

 

Let's find matrix

 

    \[ C = AB \]

 

    \[\ C = \begin{bmatrix} 3 & 5  \\ 7 & -2 \end{bmatrix} \cdot \begin{bmatrix} -4 & 6 \\ 5 & 7 \end{bmatrix} \quad \]

 

    \[\ C = \begin{bmatrix} 3. -4 + 5.5 & 3.6 + 5.7  \\ 7.-4 + -2.5 & 7.6 + -2.7 \end{bmatrix} \quad \]

 

    \[\ C = \begin{bmatrix} 13 & 53  \\ -38 & 28 \end{bmatrix} \quad \]

 

The word matrix appears five times in the King James Version, always in the phrase “all that openeth the matrix”.

Matrix is a translation of the Hebrew word רֶחֶם (rechem), meaning womb. And openeth is פֶּטֶר (peter), meaning that which first opens. So the phrase means that which first opens the womb, thus the mother’s firstborn.

The English word matrix comes from the same word in Latin, meaning womb, which comes from mater meaning motherAt the time of translation, matrix meant womb in English, but it has since taken on different meanings in biology, mathematics, computing and electronics. Because of these multiple meanings as well as its use in the Bible the Wachowskis chose the word matrix for their 1999 movie The Matrix ‒ the Matrix is both a computer simulation for the humans’ minds and a machine womb for their bodies.

We can also find the same phrase “all that openeth the womb”  in the books of Exodus, Ezekiel and Luke.

In the earlier Tyndale translation (1534) of the Bible, the word matrix appears only once in the New Testament:

23 (as that is written in the lawe of the Lorde: every man that fyrst openeth the matrix shalbe called holy to the Lorde)
(Lk. 2:23 TNT)

The word matrix in the Bible
Luke2:23 - The word matrix in the Bible
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Emma

I am passionate about travelling and currently live and work in Paris. I like to spend my time reading, gardening, running, learning languages and exploring new places.