Specialist Mathematics · Unit 1 · Matrices · Matrix arithmetic and algebra
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A researcher records the daily number of customer visits to four branches (P, Q, R, S) of a retail chain over a week. The data for Monday are: P = 342, Q = 267, R = 189, S = 421. Which matrix correctly represents these Monday visit numbers as a column vector?
A data analyst encodes information using a 3 × 4 matrix M where the entry in row i and column j is defined by m_{ij} = 2i² - 3j + 5 for i = 1, 2, 3 and j = 1, 2, 3, 4. (a) Construct matrix M. (2 marks) (b) Determine the sum of all entries in the third column of M. (1 mark) (c) A second matrix N is defined as N = M^T, where M^T denotes the transpose of M. State the order of matrix N and determine the entry n_{32}. (2 marks)
A wildlife biologist tracks three species of migratory birds (Species P, Q, and R) observed at a coastal wetland over five survey periods. The matrix \(\mathbf{B}\) records the number of individuals of each species observed during each period. (a) State the order of matrix \(\mathbf{B}\). (1 mark) (b) Identify the element \(b_{32}\) and explain what it represents in this context. (1 mark) (c) Write the matrix element that represents the number of Species Q observed in Period 4. (1 mark)
Consider the matrices A = [3 -1 2; 0 4 -3] and B = [x 5; 2y -4; 1 z]. (a) State the order of matrix A and the order of matrix B. (b) Given that the product AB exists, determine whether the product BA exists. Justify your answer. (c) Calculate the element in row 2, column 1 of the matrix product AB in terms of x, y and z. (d) Given that AB = [3x + 2 2z + 19; 8y - 3 -13] and x = 2, determine the values of y and z.
A wildlife conservation project tracks three age classes of a native marsupial species. The population vector at the start of Year 3 is given by \(\mathbf{P}_3 = \begin{bmatrix} 245 \\ 182 \\ 119 \end{bmatrix}\), where the entries represent juveniles (0–1 year), sub-adults (1–2 years), and adults (2+ years) respectively. (a) State the order of matrix \(\mathbf{P}_3\). (b) Identify the entry \(p_{23}\) in the population matrix \(\mathbf{P} = \begin{bmatrix} 230 & 195 & 245 \\ 168 & 154 & 182 \\ 112 & 107 & 119 \end{bmatrix}\), where columns represent Years 1, 2, and 3. (c) Using correct matrix notation, write the population vector that would represent a scenario where there are 310 juveniles, 0 sub-adults, and 205 adults.