A matrix is an array of numbers arranged in rows and columns. Matrices are used to represent and solve linear equations, perform geometric transformations efficiently.
An general m x n matrix has m rows and n columns and looks like:
The following matrix has 2 Rows and 3 Columns:
This is a square matrix (m = n) with ones on the diagonal and zeros elsewhere.
Example of a 4x4 identiry matrix:
The two matrices must be the same size, i.e. the rows must match in size, and the columns must match in size.
- m1n1: 3 + 4 = 7
- m1n2: 8 + 0 = 8
- m2n1: 4 + 1 = 4
- m2n2: 6 + (-9) = -3
Note
Example: a matrix with 3 rows and 5 columns can be added to another matrix of 3 rows and 5 columns.
But it could not be added to a matrix with 3 rows and 4 columns (the columns don't match in size)
To subtract two matrices: subtract the numbers in the matching positions:
- m1n1: 3 - 4 = -1
- m1n2: 8 - 0 = 8
- m2n1: 4 - 1 = 3
- m2n2: 6 - (-9) = 15
In scalar multiplication, each element of the matrix is multiplied by a scalar value.
Matrix multiplication combines two matrices to produce a new matrix.
The number of columns in the first matrix must equal the number of rows in the second matrix. So A(mxn) and B(mxn) can be multiplied when An = Bm.
- We can multiply
- 2x2 with 2x2
- 3x3 with 2x2
- 2x3 with 3x2
- We cannot multiply
- 1x4 with 1x4
- 5x6 with 9x6
- 3x5 with 3x2
Here is a theoretical example:
- m1n1: 17 + 29 + 3*11 = 58
- m1n2: 18 + 210 + 3*12 = 64
- m2n1: 47 + 59 + 6*11 = 139
- m2n2: 48 + 510 + 6*12 = 154
By organizing prices and quantities into matrices, we can efficiently calculate the total sales over multiple days using matrix multiplication.
The transpose of a matrix A is denoted as Aᵗ and is obtained by flipping the matrix over its diagonal. Rows become columns and columns become rows.
The determinant is a scalar value that can be calculated from a square matrix (where m=n). It provides important information about the matrix, such as whether it is invertible, and is used in solving systems of linear equations, among other applications.
Caution
A square matrix is invertible if and only if its determinant is not zero. If the determinant is zero, the matrix does not have an inverse.
We can write "determinant of A" as |A|, so by using the vertical bars or by using det(A). In the images below we will use both. But in most places you will see vertical bars.
It is easy to remember when you think of a cross:
- Blue is positive (+ad)
- Red is negative (−bc)
For a 3x3 matrix, the determinant is calculated using a more extended formula. It may look complicated, but there is a pattern:
- Multiply a by the determinant of the 2×2 matrix that is not in a's row or column.
- Likewise for b, and for c
- Sum them up, but remember the minus in front of the b
The pattern continues for 4×4 matrices:
- plus a times the determinant of the matrix that is not in a's row or column,
- minus b times the determinant of the matrix that is not in b's row or column,
- plus c times the determinant of the matrix that is not in c's row or column,
- minus d times the determinant of the matrix that is not in d's row or column,
Notice the +−+− pattern (+a... −b... +c... −d...). This is important to remember.
Just like the inverse of the number 5, which is 1/5 (can be written as 5⁻¹), matrixes also have inverses. When you multiply a number by its inverse you get 1. For example 5 x 5⁻¹ = 1.
The inverse of a matrix A is denoted as A⁻¹ and is the matrix that, when multiplied by A, results in the identity matrix. So A x A⁻¹ = I.
Caution
Not all matrices have inverses. A matrix must be square (same number of rows and columns) and have a non-zero determinant to be invertible.
For a 2x2 matrix, the inverse can be calculated using a straightforward formula:
Note
Remember that ad-bc is the determinant of a matrix.
Applied example:
Calculating the inverse of a 3x3 matrix is more complex and involves the following steps:
- Compute the matrix of minors.
- Turn the matrix of minors into a matrix of cofactors.
- Find the adjugate (transpose of the cofactor matrix).
- Divide by the determinant of the original matrix.
For each element of the matrix:
- Ignore the values on the current row and column
- Calculate the determinant of the remaining values
Note
The above image only shows 4 values being calculated, but we need to calulcate the determinant of all 3x3 elements.
The final result of the MoM looks as follows:
Just apply a "checkerboard" of minuses to the "Matrix of Minors". In other words, we need to change the sign of alternate cells, like this:
Now "Transpose" all elements of the previous matrix... in other words swap their positions over the diagonal (the diagonal stays the same):
And we are done! Now we have an inverse of our matrix!
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Computer Graphics: Used for 2D and 3D transformations, such as scaling, rotating, and translating objects in video games and simulations.
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Machine Learning: Used in algorithms for data representation, training neural networks, and dimensionality reduction.
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Computer Vision: Analyze and manipulate images and video data for object detection and recognition.