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.

Use Cramers rule to solve 2x-y+2z=6x-y+z=2x+y+2z=9.

Explain what it means in terms of an inverse for a matrix to have a 0 determinant.

cuny. F2 osnovski_h14b / ma119-hsi-08-applications_of_quadratic_equations / 3 MAA Previous Problem Problem List Next Problem (1 point) MA119-HSI-08-Applications of Quadratic Equations: Proble One number is 8 less than twice a second number. Find a pair of such numbers so that their product is as small as possible. These two numbers are (Use a comma to separate your numbers.) MATHEMATICAL ASSOCIATION OF AMERICA Note: You can earn partial credit on this problem. # 3 The smallest possible product is Preview My Answers Submit Answers You have attempted this problem 0 times. You have unlimited attempts remaining. Email Instructor meeting-898180....ics F3 ㅁㅁ 13 E 000 000 $ $1 4 R F4 % 5 T F5 ^ 6 MacBook Air F6 Y & 7 F7 U * 8 ▶11 F8 9 F9 O 0 F10 P -

Find the values of x and y for which (x + y, 0) = (3,1 - y)

Partially differentiate the following equations with respect to x1 and x2

Find two linearly independent solutions of y" + 5xy = 0 of the form y1 = 1+ a3a + a6x® + ... y2 = x + b,x* + byz7 +... Enter the first few coefficients: az = a6 b4 by = || ||

1) Solve for x, y and z x + 2y - z = 0 2x + 3y - 2z = 3 -x - 4y + 3z = - 2

Y₁ = 1+ a3x³ + A6㺠+... Y₂ = x + b₁x¹ +b7x7 +... Enter the first few coefficients: Az = a6 = b₁ b7 ) Find two linearly independent solutions of y" + 6xy = = - 0 of the form

1) Identify what type of equation (linear, quadratic or exponentinal) was solved in part b and c which screenshot has been attached down below and justify the choice.