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IG.M15 Matrices: Solving Simultaneous Equations With the Inverse Matrix

EDEXCEL 4MA1 · Calculator allowed · about 45 minutes

Total Marks

Name: _______________________________ Date: ____ / ____ / ______

Answer ALL questions. Show all your working.

Write down the 2x2 identity matrix I.

(Total for Question 1 is 1 mark)

Given matrix M = [[2, 1], [1, 1]] and vector x = [p, q]^T, write the matrix equation M x = c that represents the simultaneous equations 2p + q = 9 and p + q = 5. State c explicitly.

(Total for Question 2 is 1 mark)

For M = [[3, 2], [1, 1]], calculate det(M). This determinant is used when forming M^-1.

(Total for Question 3 is 2 marks)

Given M = [[4, 1], [3, 2]] and c = [14, 16]^T, a student claims x = M^-1 c can be used to solve for x. Write down the first multiplication step that must be done to compute x, showing M^-1 explicitly in terms of 1/det and the adjugate (do not compute numeric entries).

(Total for Question 4 is 2 marks)

M = [[2, 3], [5, 4]] and c = [9, 19]^T. Use a calculator to find x = M^-1 c and hence give the solution (x1, x2) for the simultaneous equations 2x1 + 3x2 = 9 and 5x1 + 4x2 = 19. Give answers as integers.

(Total for Question 5 is 3 marks)

Write down the condition on det(M) for a 2x2 matrix M to be invertible, and state why this is needed when solving M x = c.

(Total for Question 6 is 3 marks)

A small cafe sells two types of sandwich: cheese and ham. In one day the cafe sells 40 sandwiches in total. Each cheese sandwich uses 2 slices of cheese and 1 slice of tomato. Each ham sandwich uses 1 slice of cheese and 2 slices of tomato. That day the kitchen used 62 slices of cheese and 58 slices of tomato. Form two simultaneous equations, write them in matrix form M x = c, and use the inverse matrix method to find how many cheese and ham sandwiches were sold. Show your working and check your values by substitution.

(Total for Question 7 is 5 marks)

Translate the following word problem into matrix form and solve by the inverse matrix method. A florist arranges bouquets with roses and lilies. Each red bouquet uses 3 roses and 2 lilies. Each white bouquet uses 2 roses and 4 lilies. The florist made 17 bouquets in total and used 44 roses. Form the simultaneous equations, write M x = c, and find how many red and white bouquets were made. Include a check.

(Total for Question 8 is 5 marks)

Solve the simultaneous equations 4a + 5b = 41 and 3a + 2b = 26 using matrices. Use the inverse of the coefficient matrix and show your calculation clearly. Give the exact values for a and b and verify by substitution.

(Total for Question 9 is 5 marks)

A shop sells pencils and pens. Two pencils and three pens cost 7 pounds. Four pencils and one pen cost 6 pounds. Let p be the cost of a pencil and q the cost of a pen in pounds. Write the equations in matrix form and use the inverse matrix method to find p and q. Give answers as fractions in simplest form and include a verification.

(Total for Question 10 is 5 marks)

Consider the system x + 2y = 8 and 6x + 5y = 47. Use the inverse matrix method to find x and y. Include full calculation of M^-1 and a final substitution check.

(Total for Question 11 is 5 marks)

Given M = [[1, -1], [2, 3]] and c = [4, 11]^T, write down the matrix equation M x = c and state whether det(M) is zero or non zero.

(Total for Question 12 is 1 mark)

For M = [[7, 2], [4, 1]] and c = [20, 11]^T, use a calculator to find x = M^-1 c. Give the ordered pair solution and give a one line verification by substitution into the first original equation only.

(Total for Question 13 is 1 mark)

State one advantage of using the inverse matrix method over substitution when solving two linear equations that come from a word problem, giving a brief reason.

(Total for Question 14 is 1 mark)

Mark scheme · IG.M15 Matrices: Solving Simultaneous Equations With the Inverse Matrix

Question 1

B1 I = [[1, 0], [0, 1]] cao
Answer: I = [[1, 0], [0, 1]]

Question 2

B1 M x = c with c = [9, 5]^T cao
Answer: M x = [ [2,1],[1,1] ] [p, q]^T = [9, 5]^T

Question 3

M1 writes det(M) = 3*1 - 2*1 or equivalent
A1 det(M) = 1 cao
Answer: det(M) = 1

Question 4

M1 writes M^-1 = 1/det(M) * [[2, -1], [-3, 4]] or equivalent adjugate form
A1 shows x = (1/det(M)) [[2, -1], [-3, 4]] [14,16]^T cao
Answer: x = (1/det(M)) [[2, -1], [-3, 4]] [14, 16]^T

Question 5

M1 uses inverse method: forms M^-1 and begins multiplication with c, or solves by showing correct matrix multiplication set-up
M1 carries out multiplication to obtain numerical vector, correct up to a factor of 1/det if applicable
A1 gives (x1, x2) = (3, 1) cao
Answer: (x1, x2) = (3, 1)

Question 6

B1 det(M) not equal to 0 stated
B1 reason: inverse matrix M^-1 exists only when det(M) not equal to 0
B1 reason: without M^-1 you cannot form x = M^-1 c to find a unique solution, oe
Answer: det(M) must not equal 0, because only then does M^-1 exist and x = M^-1 c gives the unique solution.

Question 7

Level 1 (1-2): Forms at least one correct equation from the worded context and shows partial matrix set-up
Level 2 (3-4): Correctly forms both simultaneous equations, writes M x = c, and applies M^-1 to c with some correct arithmetic, producing a candidate solution
Level 3 (5): Gives fully correct equations and matrix form, computes M^-1 correctly, finds integer values for both unknowns using x = M^-1 c, and verifies both equations by substitution
Indicative content:
- Let x = number of cheese sandwiches, y = number of ham sandwiches
- Equations: x + y = 40 and 2x + y = 62 for cheese slices, or use tomato equation 1x + 2y = 58; any two independent equations are acceptable
- Matrix form example: [[1,1],[2,1]] [x,y]^T = [40,62]^T or using tomato [40,58]^T; use a consistent pair
- Compute det = 1*1 - 1*2 = -1 and inverse M^-1 = -1 * [[1,-1],[-2,1]] = [[-1,1],[2,-1]] if using the first matrix, or show correct numeric inverse for the chosen matrix
- Multiply M^-1 by c to obtain [x,y]^T = [22,18]^T (or equivalent correct pair), and substitute into 2x + y and x + 2y to check totals 62 and 58

Question 8

Level 1 (1-2): Forms one correct equation from the context or writes a matrix with one row correct
Level 2 (3-4): Forms both simultaneous equations, writes M x = c, and applies inverse with partly correct arithmetic
Level 3 (5): Correct matrix equation, computes M^-1 correctly, obtains integer solution and substitutes back to verify both original equations
Indicative content:
- Let r = red bouquets, w = white bouquets
- Equations: 3r + 2w = 44 and r + w = 17
- Matrix form M = [[3,2],[1,1]], x = [r,w]^T, c = [44,17]^T
- det = 3*1 - 2*1 = 1 so M^-1 = [[1, -2], [-1, 3]]
- x = M^-1 c = [[1,-2],[-1,3]] [44,17]^T = [44 -34, -44 +51]^T = [10,7]^T, check 3*10 + 2*7 = 44 and 10 + 7 = 17

Question 9

Level 1 (1-2): Shows the matrix form or computes det or adjugate partially
Level 2 (3-4): Forms M x = c and computes M^-1 with some arithmetic, leading to a candidate solution
Level 3 (5): Correct M^-1, correct multiplication giving a and b, and successful substitution into both equations
Indicative content:
- M = [[4,5],[3,2]], x = [a,b]^T, c = [41,26]^T
- det = 4*2 - 5*3 = 8 - 15 = -7, adjugate = [[2, -5], [-3, 4]] so M^-1 = -1/7 [[2,-5],[-3,4]]
- Multiply adjugate by c: [[2,-5],[-3,4]] [41,26]^T = [82 -130, -123 +104] = [-48, -19]
- Multiply by -1/7 gives [48/7, 19/7], so a = 48/7, b = 19/7, and check by substitution into original equations

Question 10

Level 1 (1-2): Forms one equation correctly or writes matrix form with one row incorrect
Level 2 (3-4): Forms both equations, computes inverse with partial accuracy, and gets close to the correct fractional answers
Level 3 (5): Correct M^-1 used, p and q found as simplified fractions, and substitution verifies both equations
Indicative content:
- Equations: 2p + 3q = 7 and 4p + q = 6
- Matrix M = [[2,3],[4,1]], x = [p,q]^T, c = [7,6]^T
- det = 2*1 - 3*4 = 2 - 12 = -10, adjugate = [[1,-3],[-4,2]]
- M^-1 = -1/10 [[1,-3],[-4,2]]
- Multiply and simplify to p = 11/10 (1.10) and q = 8/5 (1.60); check 2*(11/10) + 3*(8/5) = 2.2 + 4.8 = 7 and 4*(11/10) + 8/5 = 4.4 + 1.6 = 6, both correct

Question 11

Level 1 (1-2): Forms M x = c and shows some attempt at inverse computation
Level 2 (3-4): Computes M^-1 with minor arithmetic slip or obtains one variable correctly
Level 3 (5): Correct inverse, correct multiplication giving integer x and y, and both equations verified by substitution
Indicative content:
- M = [[1,2],[6,5]], x = [x,y]^T, c = [8,47]^T
- det = 1*5 - 2*6 = 5 - 12 = -7, adjugate = [[5, -2], [-6, 1]]
- M^-1 = -1/7 [[5,-2],[-6,1]]
- Multiply adjugate by c: [[5,-2],[-6,1]] [8,47]^T = [40 -94, -48 +47] = [-54, -1], multiply by -1/7 gives [54/7, 1/7], so x = 54/7, y = 1/7, and check in both original equations

Question 12

B1 M x = c written correctly and det(M) = 1*3 - (-1)*2 = 3 +2 = 5 stated non zero
Answer: M x = [[1,-1],[2,3]] [x]^T = [4,11]^T and det(M) = 5, non zero

Question 13

B1 correct pair x = (2,3) and verifies 7*2 + 2*3 = 20; accept correct solution and valid substitution
Answer: (x1, x2) = (2,3); check: 7*2 + 2*3 = 14 + 6 = 20

Question 14

B1 advantage stated, e.g. inverse method is systematic and faster for multiple similar problems, oe, or gives both unknowns in one calculation, with brief reason
Answer: The inverse matrix method gives both unknowns in one calculation, which is faster and less error prone for repeated calculations or when coefficients are not simple.

Matrices: Solving Simultaneous Equations With the Inverse Matrix - Worksheets, Questions and Revision