MAT1503 EXAM PACK 2025 (PRACTICE MATERIALS WITH COMPLETE SOLUTIONS)

MAT1503 EXAM PACK 2025 (PRACTICE MATERIALS WITH COMPLETE SOLUTIONS) Gaussian Elimination - answers1) Put the matrix in augmented matrix form 2) Use row operations to put the matrix in echelon form 3) Write the equations from the echelon form matrix 4) Solve the equations. Gauss-Jordan Elimination - answers1) Put the matrix in augmented matrix form 2) Use row operations to put the matrix in reduced echelon form 3) Write the equations from the echelon form matrix 4) Solve the equations. trivial solution - answersThe solutions of the homogeneous linear systems are 0 non-trivial solution - answersThe solutions of the homogeneous linear systems are infinite (free variables are used) Free Variable Theorem for Homogeneous Systems - answersIf a homogeneous linear system has n unknowns, and its augmented matrix has r nonzero rows in reduced row echelon form, then the system has n - r free variables entries - answersThe numbers in the matrix Theorem 1.2.2 - answersA homogeneous linear system with more unknowns than equations has infinitely many solutions back-substitution - answers1) Solve the equations for the leading variables 2) Substitute each equation into all equations above it, starting at the bottom 3) Assign arbitrary values to any free variables column vector - answersA matrix with only one column row vector - answersA matrix with only one row scalars - answersNumerical quantities square matrix of n - answersA matrix with n rows and n columns main diagonal - answersThe line of entries between the first entry in the first row and the last entry in the last row equal matrices - answersTwo matrices of the same size with matching corresponding entries addition of matrices - answersThe entries of one matrix are added to the corresponding entries of another matrix of the same size subtraction of matrices - answersThe entries of one matrix are subtracted from the corresponding entries of another matrix of the same size scalar multiple - answersObtained by multiplying each entry of a matrix with a scalar multiplication of matrices - answers1) Multiply the first row of the first matrix with corresponding entries in the first column of the second matrix 2) Add up the resulting products to find the new entry 3) Continue until every row of the first matrix is multiplied with every column of the second matrix commute - answersIf AB = BA. This does not always happen submatrix - answersPartitioned from a matrix by adding horizontal and vertical rules between selected rows and columns linear combination - answersIf A₁, A₂, ... , A are matrices of the same size, and if c₁, c₂, ... , c are scalars, then it can be expressed in the form c₁A₁ + c₂A₂ + ... + cA Theorem 1.3.1 - answersIf A is an m x n matrix, and if x is a column vector, the product Ax can be expressed as a linear combination of the column vectors of A in which the coefficients are the entries of x transpose matrix - answersObtained by interchanging the rows and columns of a matrix reflecting a matrix - answersInterchanging entries that are symmetrically positioned about the main diagonal of a square matrix trace of a matrix - answersThe sum of the entries on the main diagonal of a square matrix Properties of Matrix Arithmetic - answersa) A + B = B + A b) A + (B + C) = (A + B) + C c) A(BC) = (AB)C d) A(B + C) = AB + AC e) (B + C)A = BA + BC f) A(B - C) = AB - AC g) (B - C)A = BA - BC h) a(B + C) = aB + aC i) a(B - C) = aB - aC j) (a + b)C = aC + bC k) (a - b)C = aC - bC l) a(bC) = (ab)C m) a(BC) = (aB)C / B(aC) Properties of Zero Matrices - answersa) A + 0 = 0 + A = A b) A - 0 = A c) A - A = A + (-A) = 0 d) 0A = 0 e) If cA = 0, then c = 0 or A = 0 Theorem 1.4.3 - answersIf R is the reduced row echelon form of an n × n matrix, then either R has a row of zeros or R is the identity matrix Definition 1 - answersIf A is a square matrix, and if a matrix B of the same size can be found such that AB = BA = I, then A is said to be invertible and B is called the inverse Theorem 1.4.4 - answersIf B and C are both inverses of the matrix A, then B = C inverse of an invertible matrix - answersA×A⁻¹ = I and A⁻¹A = I Theorem 1.4.5 - answersA matrix is invertible if and only if the determinant ≠ 0, in which case the inverse is given by the formula A⁻¹ = 1/det(A) determinant - answersThe difference between the two diagonals in a 2×2 matrix. Theorem 1.4.6 - answersIf A and B are invertible matrices with the same size, then AB is invertible and (AB)⁻¹ = B⁻¹A⁻¹ A product of any number of invertible matrices is invertible, and the inverse of the product is the product of the inverses in reverse order Theorem 1.4.7 - answersIf A is invertible and n is a nonnegative integer, then a) A⁻¹ is invertible and (A⁻¹)⁻¹ = A b) Aⁿ is invertible and (Aⁿ)⁻¹ = A⁻ⁿ = (A⁻¹)ⁿ c) kA is invertible for any nonzero scalar k, and (kA)⁻¹ = k⁻¹A⁻₁ Properties of the Transpose - answersIf the sizes of the matrices are such that stated operations can be performed, then a) (Aᵀ)ᵀ = A b) (A + B)ᵀ = Aᵀ + Bᵀ c) (A - B)ᵀ = Aᵀ - Bᵀ