Exercise practice: Matrices

4.1 Exercise 3.1: orders and entries

The first exercise builds the vocabulary. Typical questions ask for the order of a matrix, the number of entries, or the value of one entry given a formula. For a matrix with entries aij = i plus 2j of order 2 by 3, the entry a23 is row 2, column 3, so 2 plus 6 equals 8. A question may also give the number of entries and ask for possible orders, which comes from the factor pairs of that count.

Equality questions come next. If two matrices of the same order are equal, corresponding entries are equal, and those equations determine the missing numbers. Write the equations in order and solve them as a small linear system.

4.2 Exercise 3.2 and 3.3: operations and products

Exercise 3.2 is arithmetic with matrices. Add or subtract matching entries, multiply every entry by the scalar, and check the order has not changed. Exercise 3.3 is where the order rule matters. A worked example: if A has order 2 by 3 and B has order 3 by 2, then AB has order 2 by 2 while BA has order 3 by 3, so the two are different sizes.

For a product question, compute one entry at a time and write the row and column used. For A = [[1, 2], [3, 4]] and B = [[2, 0], [1, 2]], the top left entry of AB is 1 times 2 plus 2 times 1, which is 4, and the top right entry is 1 times 0 plus 2 times 2, which is 4. Continue until the full product is assembled.

4.3 Exercise 3.4 and Miscellaneous: transpose and symmetry

Exercise 3.4 asks for transposes, symmetry checks and decompositions. The key test is to compute the transpose and compare it with the original. For a matrix to be symmetric, the entry in row i and column j must equal the entry in row j and column i for every pair. For skew symmetric, those entries must be negatives, which forces every diagonal entry to be zero.

The Miscellaneous Exercise mixes the ideas. A standard item asks for a matrix A such that A plus its transpose is symmetric, or for the symmetric and skew symmetric parts of a given matrix. Another item uses the identity matrix to solve a matrix equation, which reduces to adding and multiplying matrices in the usual way.

4.4 How to present the answer

For a product, show the row and column multiplication once, then write the final matrix. For an equality question, list each equation. For a symmetry question, compute the transpose explicitly and compare entries. For a decomposition, state both parts and add them back as a check. This habit makes the answer easy to mark.