Matrices and determinants

Systems of linear equations are solved using matrices and determinants. Matrices are used to organize data in rows or columns, while determinants help check the invertibility of square matrices and determine if their solutions exist.

A matrix is a rectangular collection of numbers arranged in rows and columns. These numbers are referred to as elements. Meanwhile, a determinant is a single value calculated using a square matrix.

The determinant of a matrix provides key information about the matrix, including whether it has an inverse and how it transforms space.

Matrices and determinants are often associated with each other, but are significantly different from each other in the following ways:

When working with matrices, we should know some important properties of matrices that are useful in matrix operations.

• Addition and subtraction: Addition and subtraction of matrices are only possible if the matrices are of the same order. 
   
• Scalar multiplication: Each element in a matrix can be scaled by multiplying by a scalar.
   
• Matrix multiplication: The product is defined only when the number of columns in the first equals the number of rows in the second.
   
• Transpose: The transpose of a matrix is formed by switching the positions of rows with columns across the main diagonal.
   
• Symmetric and skew-symmetric matrices: A matrix is called symmetric if it is equal to its transpose (A = AT). A matrix is called skew-symmetric if it is the negative of its transpose (A = - AT).
   
• Identity matrix: The identity matrix I serves as a multiplicative identity. The diagonal elements are always 1, and the others are zero.
   
• Inverse matrix: The inverse of a matrix exists only for square matrices with a non-zero determinant.
   
• Eigenvalues: Matrices have eigenvalues, which can be found using the characteristic polynomial.
   
• Trace: The sum of a matrix’s diagonal elements gives the trace of that matrix.

Understanding how to solve problems using matrices and determinants helps deal with systems of linear equations.

Matrices and determinants organize complex data and simplify it.

Depending on the matrix, we can choose the most suitable methods to solve for them.

Any linear equation Ax + By = C is written as AX = B in matrix form.
Where A is the coefficient matrix, X is the column matrix of variables, and B is the constant matrix.

Such an equation can be solved using the following methods:

Method 1: Inverse matrix method

If a matrix is non-singular and invertible, we use the inverse matrix method to find its solution. We can find the solution using X = A-1B.

Step 1 - Find the inverse of matrix A

Step 2 - Multiply the inverse by matrix B

The result containing the values is the solution matrix X.
This method is most suitable for small square matrices of order 2 × 2 or 3 × 3 because of computational complexity.

Method 2: Gauss Elimination Method

The Gauss elimination method simplifies a matrix one step at a time using elementary row operations. 

Step 1 - Convert the augmented matrix [A|B] using row operations.

Step 2 - Get the zeroes below the diagonal to form an upper triangular matrix.

Step 3 - Use back-substitution to solve the resulting system.
This method is more preferable for larger systems.

Method 3: Gauss-Jordan Elimination

This method is an extension of Gaussian elimination and reduces the matrix to reduced row-echelon form (diagonal form). This removes the need for back substitution.

Step 1 - Perform row operations till the identity matrix is found on the left side of the augmented matrix.

Step 2 - The right side is the solution.
This method gives a direct solution and is useful in programming and numerical software.

Cramer’s Rule is a determinant-based method used to solve systems of n linear equations in n variables for square systems, provided that the coefficient matrix A is square and det(A)  0.

For a given system; 
a1x + b1y = c1  
a2x + b2y = c2
Its matrix form is written as AX = B
Where A is the coefficient matrix, X is the variable matrix, and B is the constant matrix.

Follow these steps to apply Cramer’s rule:        

Step 1 - Find the determinant of the coefficient matrix det(A).

Step 2 - Replace the first column of A with B and find the new determinant, Dx.

Step 3 - Replace the second column of A with B and find Dy.

Step 4 - Apply the formulas, x=Dxdet(A), y =Dydet(A)
Where det(A) is the determinant of the coefficient matrix, Dx and Dy are the determinants formed by replacing columns x and y with constants.

Cramer’s rule is useful for smaller systems.

When we simplify a determinant, the matrix is reduced to a single number or scalar value. To find these values, we can compute the determinant in the following ways:

For a 2 × 2 matrix A

\(\begin{bmatrix} a & b \\ c & d \end{bmatrix} \)

The determinant is det(A) = ad - bc
 

To find a 3 × 3 determinant 
Let matrix A,

\(A = \begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix}\)

Using cofactor expansion along the first row, det(A) = a(ei-fh)-b(di-fg)+c(dh-eg)

We can also use Sarrus’ rule to calculate the determinant of a 3 × 3 matrix. It is considered a shortcut method and is limited only to 3 × 3 matrices.

Follow these two steps to use the Sarrus’ rule:

Step 1 - Repeat the first two columns next to the matrix.

Step 2 - Multiply diagonals from top-left to bottom-right and subtract the sum of diagonals from bottom-left to top-right.

There are two conditions of solvability to keep in mind while solving for matrices and determinants:

• Unique solution: When det(A)  0, the system has a unique solution.
   
• No solution or infinite solutions: If det(A) = 0, then the system is either inconsistent or dependent.

Matrices and determinants are important in solving complex systems across various fields. Some of their applications are listed below.

• Structural load analysis in civil engineering: Determinants help solve systems of equations in structural analysis to look for solutions that may be unique, consistent, or unstable in force and stress distribution.
   
• Secure message encryption using the Hill cipher: Encryption algorithms like the Hill cipher depend on invertible matrices for encrypting and decrypting messages in modular arithmetic.
   
• Economic Forecasting Using the Leontief Input-Output Matrix: Economists use the Leontief Input-Output Model, which uses matrices to forecast economic output and plan resources. For
  uch extra raw material and labor will be required from other sectors.
   
• Robotic movement in industrial automation: Matrices are used in robotics for the calculation of the position and orientation of robotic arms through kinematics.
   
• Image compression in digital signal processing: Image data is stored as matrices, and techniques like singular value decomposition (SVD) use matrix factorization to reduce file sizes while preserving essential information in images.