One cannot fail to notice that in forming linear combinations of linear equations there is no need to continue writing the 'unknowns' x1,...,x, since one actually computes only with the coefficients A,, and the scalars y. We shall now abbreviate the system (1-1) by Do not solve using AI, I want real solutions with graphs and codes, wherever required. Reference is given, if you need further use hoffmann book of LA or maybe Friedberg. where AX = Y An AL A Aml 1 and Y -N We call A the matrix of coefficients of the system. Strictly speaking, the rectangular array displayed above is not a matrix, but is a repre- sentation of a matrix. An m X n matrix over the field F is a function A from the set of pairs of integers (i, j), 1≤i≤m, 1≤ j ≤ n, into the field F. The entries of the matrix A are the scalars A(i,j) = A, and quite often it is most convenient to describe the matrix by displaying its entries in a rectangular array having m rows and n columns, as above. Thus X (above) is, or defines, an n XI matrix and Y is an m × I matrix. For the time being, AX = Y is nothing more than a shorthand notation for our system of linear equations. Later, when we have defined a multi- plication for matrices, it will mean that Y is the product of A and X. We wish now to consider operations on the rows of the matrix A which correspond to forming linear combinations of the equations in the system AX=Y. We restrict our attention to three elementary row operations on an m Xn matrix A over the field F: 1. multiplication of one row of A by a non-zero scalar e; 2. replacement of the rth row of A by row r plus c times row 3, c any scalar and r8; 3. interchange of two rows of A. Problem 2: The Invertibility and Uniqueness of Matrix Inverses Statement: Let A be an nxn invertible matrix. The inverse of A, denoted by A¹, satisfies the conditions AA and AA = I, where I is the identity matrix of the same order. Tasks: 1. Prove that if A is invertible, then the inverse is unique. Use a general argument based on the properties of matrix multiplication and the definition of the identity matrix. 2. Show that if A and B are n x n matrices such that AB = I, then both A and B must be invertible and B = A-. Provide a theoretical justification using properties of matrix multiplication and inverses. 3. Graphically interpret the concept of matrix inversion in terms of transformations in R". Explain how an invertible matrix represents a transformation that maps vectors uniquely back to their original positions. Use a diagram to illustrate this concept. 4. Discuss the relationship between the invertibility of a matrix and its determinant. Prove that a matrix is invertible if and only if its determinant is non-zero, using the properties of row operations and the determinant function.