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Unit 13: Systems of Equations and Matrices

Solving several equations at once by elimination and by row-reducing a matrix, and using determinants and inverses to solve systems directly.

8 topics · 31 questions

Each topic is a path, not a menu: work down it in order and every question builds on the one above. Click any of them to watch the answer draw itself.

Solving Linear Systems by Substitution and Elimination

  1. How it worksHow does eliminating one variable by adding or subtracting equations reduce a system to a single equation?
  2. Worked exampleFind the solution to the system 2x + 3y = 12 and x - y = 1.
  3. Common mistakeWhy doesn't a system of two linear equations always have exactly one solution?
Your turn1 of 2
Find x in the system 4x + 4y = 28, x - y = d, as a function of d.

Systems of Nonlinear Equations

  1. How it worksHow does substitution turn a linear-quadratic system into a single equation you can solve directly?
  2. Worked exampleFind the intersection points of the line y = x + 1 and the circle x2 + y2 = 25.
  3. Common mistakeWhy can't you just add and subtract equations to solve a nonlinear system the way you would with two lines?
  4. Ties togetherHow does substituting into a nonlinear system reduce to the same elimination ideas used for linear systems?
Your turn1 of 2
Find the larger x-coordinate of the intersection of the line y = x + b and the circle x2 + y2 = 132, as a function of b.

Matrices, Addition, and Scalar Multiplication

  1. How it worksHow do you add two matrices or multiply a matrix by a scalar, entry by entry?
  2. Worked exampleFind A + B and 2A for A = [[1,2],[3,4]] and B = [[5,-1],[0,2]].
  3. Common mistakeWhy can't you add two matrices that have different dimensions?
Your turn1 of 2
Matrix A has (1,1) entry 4. Matrix B has (1,1) entry x. Find the (1,1) entry of A + B, as a function of x.

Matrix Multiplication

  1. How it worksHow does each entry of a matrix product come from a row of one matrix dotted with a column of the other?
  2. Worked exampleFind the product AB for A = [[1,2],[3,4]] and B = [[2,0],[1,3]].
  3. Worked exampleFind the product BA for the same matrices A = [[1,2],[3,4]] and B = [[2,0],[1,3]], and compare it with AB.
  4. Common mistakeWhy doesn't AB usually equal BA even when both products are defined?
  5. Going furtherWhat happens to matrix multiplication when the number of columns in the first matrix doesn't match the rows in the second?
Your turn1 of 2
Matrix A has first row [4, 3] and second row starting with 8. Matrix B has first column [x, -4]. Find the (1,1) entry of the product AB, as a function of x.

Solving Systems with Row Reduction

  1. How it worksHow does Gaussian elimination turn an augmented matrix into row-echelon form to reveal a system's solution?
  2. Worked exampleFind the solution to the system x + 2y - z = 3, 2x - y + z = 1, x + y + 2z = 8 using row reduction.
  3. Common mistakeWhy doesn't a row of all zeros in row-echelon form automatically mean a system has no solution?
  4. Common mistakeWhy doesn't reaching row-echelon form always mean a system ends up with exactly one solution?
Your turn1 of 2
Find z in the system x + 2y - z = 7, 2x - y + z = 7, x + y + 2z = 9 using row reduction.

The Inverse of a Matrix

  1. How it worksHow does the formula for a 2 by 2 inverse use the determinant to flip and rescale the matrix's entries?
  2. Worked exampleFind the inverse of the matrix [[3,1],[5,2]].
  3. Common mistakeWhy doesn't every square matrix have an inverse?
  4. Ties togetherHow does multiplying both sides of a matrix equation by an inverse solve a system the same way row reduction does?
Your turn1 of 2
Find the (1,1) entry of the inverse of the matrix [[a,2],[2,9]], as a function of a.

Determinants and Cramer's Rule

  1. How it worksHow does Cramer's Rule replace a column of the coefficient matrix with the constants to solve for each variable?
  2. Worked exampleFind x and y in the system 3x + 2y = 7, x - y = 1 using Cramer's Rule.
  3. Worked exampleFind z in the system x + y + z = 6, 2x - y + z = 3, x + 2y - z = 2 using Cramer's Rule.
  4. Common mistakeWhy doesn't Cramer's Rule work when the determinant of the coefficient matrix equals zero?
Your turn1 of 2
Find x in the system 4x + 4y = 19, x - y = 2 using Cramer's Rule.

Partial Fraction Decomposition

  1. How it worksHow does matching numerators after clearing denominators solve for each unknown constant in a partial fraction decomposition?
  2. Worked exampleFind the partial fraction decomposition of (3x+5)/((x-1)(x+2)).
  3. Common mistakeWhy does a repeated linear factor in the denominator need two separate terms in its partial fraction decomposition?
  4. Ties togetherHow does solving for the unknown constants in a decomposition become its own small system of linear equations?
Your turn1 of 2
Find the partial fraction decomposition of (4x+6)/((x-6)(x+4)), writing A/(x-a) + B/(x+b) as a single expression in x.