What is Matrix? Inverse Matrix Formula | Adjoint & Covariance Matrix

What is Matrix?

A matrix is a popular term in mathematics that is defined as the array of functions or numbers. Each matrix has two dimensions in the form of rows and columns. The number of rows is shown with the alphabet m and the number of columns is shown through the alphabet n. There are a plenty of parameters to categorize a matrix like the value of elements, the order of a matrix, and the total number of rows and columns etc.

What is Matrix?

With the help of matrix study in mathematics, an engineer can design buildings, they can build powerful video games, and define animations that look like 3-dimensional or 4-dimensional, and many more applications. They are also used to solve complex linear equations, in this case, calculations are completed through the computer but a depth understanding of matrix formulas are still necessary.

Matrix Formula

Matrix is an ordered arrangement of array or variable in the form of rectangular or square rows or columns. The individual number in the matrix is given as an entity or element. In case of the matrix, the numbers are written in between square brackets as shown below in the example – For example:

Addition of Matrix

If A = \begin{bmatrix}
a_{11} & a_{12}\\
a_{21} & a_{22}\\
a_{31} & a_{32}
\end{bmatrix} and B = \begin{bmatrix}
b_{11} & b_{12}\\
b_{21} & b_{22}\\
b_{31} & b_{32}
\end{bmatrix} let us calculate A + B.

Here, both matrices A and B are of same size (3 x 2).
This simplies

C = A + B = \begin{bmatrix}
a_{11} + b_{11} & a_{12} + b_{12}\\
a_{21} + b_{21} & a_{22} + b_{22}\\
a_{31} + b_{31} & a_{32} + b_{32}
\end{bmatrix}

 

Substraction of Matrix

If A = \begin{bmatrix}
a_{11} & a_{12}\\
a_{21} & a_{22}\\
a_{31} & a_{32}
\end{bmatrix} and B = \begin{bmatrix}
b_{11} & b_{12}\\
b_{21} & b_{22}\\
b_{31} & b_{32}
\end{bmatrix} let us calculate A – B.

Here, both matrices A and B are of same size (3 x 2).
This simplies

C = A – B = \begin{bmatrix}
a_{11} – b_{11} & a_{12} – b_{12}\\
a_{21} – b_{21} & a_{22} – b_{22}\\
a_{31} – b_{31} & a_{32} – b_{32}
\end{bmatrix}

 

Multiplication of Matrix

If A = \begin{bmatrix}
a_{1} & a_{2}\\
a_{3} & a_{4}\end{bmatrix} and B = \begin{bmatrix}
b_{1} & b_{2}\\
b_{3} & b_{4}\end{bmatrix} let us calculate A X B.

Here, both matrices A and B are of same size (3 x 2).
This simplies

C = A X B = \begin{bmatrix}
a_{1} b_{1} + a_{2} b_{3} & a_{1} b_{2} + a_{2} b_{4}\\
a_{3} b_{1} + a_{4} b_{3} & a_{3} b_{2} + a_{4} b_{4}\end{bmatrix}

 

The adjoint of a 2×2 matrix is given as,

If A = \begin{bmatrix}
a_{11} & a_{12}\\
a_{21} & a_{22}\end{bmatrix}

adj(A) = \begin{bmatrix}
a_{22} & -a_{12}\\
-a_{21} & a_{11}\end{bmatrix}

The horizontal lines are defined as the rows in the matrix and vertical lines are taken as the column in the matrix. For a matrix having m rows and n columns, the order of matrix is defined as m × n. for the given example, there are three rows and three columns, so the order of above matrix would be 3 × 3.Two matrices of same order can be added or subtracted by following certain rules and conditions.

The applications of matrices are common in engineering and it can be used for practicing cryptography techniques. When information is exchanged between two parties, it will stay safe due to matrix formula enablement. It is suitable for Fourier analysis, Gauss theorem, to find the electric currents, to measure the force on the bridge and many more things are completed with the help of matrix implementation.

Covariance Matrix Formula

A Covariance Matrix is a measure of how two random variables get modified altogether. It is actually needed to compute the covariance for every column in the data matrix. The other popular name for covariance matrices is dispersion matrix or variance-variance matrix. The covariance formula in mathematics is given as –

\[\LARGE Cov(X,Y)=\sum \frac{(X_{i}-\overline{X})(Y_{i}-\overline{Y})}{N}=\sum \frac{X_{i}Y_{i}}{N}\]

Where,

N = Number of scores in each set of data

X = Mean of the N scores in the first data set

Xi = ith raw score in the first set of scores

\[\LARGEx_{i}\] = $i^{th}\] deviation score in the first set of scores

Y = Mean of the N scores in the second data set

\[\LARGEY_{i}\] = $i^{th}\] raw score in the second set of scores

\[\LARGEy_{i}\] = $i^{th}\] deviation score in the second set of scores

Cov(X, Y) = Covariance of corresponding scores in the two sets of data

Where each of variables has its own meaning and significance.

Inverse Matrix Formula

The square matrix having an inverse is written as either non-singular or invertible and a square matrix whose inverse cannot be calculated is named as singular or non-invertible matrix. Keep in mind that not all square matrices have inverse and non-square matrices don’t have inverses. For the matrix A, the inverse is written as A-1.

The inverse of a 2×2 matrix is given as,

\[\large A^{-1}=\frac{1}{|A|}\times adj(A)\]

You must be thinking why we need an inverse of a matrix. Actually, we could not divide a matrix in mathematics. But this is possible to multiply a square matrix by its inverse that completes the same job.


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