In linear algebra, determinants and matrices are basic ideas. Determinants are unique numbers connected to square matrices, while matrices themselves are rectangular arrays of numbers. They have several uses in physics, computer, science, mathematics, and many other disciplines.

Let's delve deeper nto each idea : 

* Matrices 

A rectangular array of numbers arranged in rows and columns is referred to as a matrix. A capital letter is often used to indicate it. As an illustration, think about matrix A:

A=[a11 a12 a13]

[a21 a22 a23]

The members of the matrix a in this instance are a11, a12, a13, a21,a22,and a23. The row number is indicated by the first subscript, while the column number is shown by the second subscript.

Based on their size, matrices can be categorized. A matrices is referred to as a m x n matrix if it has m rows and columns. The matrix A in the example above is a 2*3 matrix.

Under specific circumstances, matrices can be  multiplied, divided, and added . The definition of matrix multiplication is as follows :

-The product AB is a m x p matrix if A is a m x n matrix and B is a n x p matrix.

- The addition and substraction of matrices are done element by element.

 * Determinants

A scalar value connected to a square matrix is a determinant Denoted by det (A) or |A|, with A standing for the matrix. Only square matrices those with an equal number of rows and columns and columns allow the definitionof the determinant.

Important details about a matrix, such as whether it is invertible or unique and the scaling factor it employs for vectors, are provided by the determinant. when a matrix's determinant is zero, the matrix is unique and lacksan inverse.

The formula for the determinant of the 2*2 matrix [a b ; c d] is ad-bc.

The determinant can be calculated for bigger matrices using a vaiety of techniques, including cofactor expansion, row operations, and determinant-specific features. The specific approach is determined by the size of the matrix and the processing resources at hand.

Determinants can be used to solve systems of linear equations locate matrices elgevaules and elgenvectors, and calculate the surface area or volume of geometric forms among other things.

These ideas serve as a foundations for more complex linear algebra topics including eigendecompositin, singular value decomposition (SVD), and matrix transformations.