Higher Mathematics 1st Paper · Classes 11-12 · Chapter 1 · 1.2

Equality, Addition, Subtraction and Scalar Multiples of Matrices.

You can tell whether two matrices can be added without reading a single entry. You only check that their orders match. If they do, the work is easy: take the two numbers in the same cell and add or subtract. Matching cell by cell also gives you the unknowns.

1. When two matrices are equal

Two matrices are equal when their orders are the same and their corresponding entries are equal. A = [[x, −4], [7, 3]] and B = [[6, −4], [7, 3]] have the same order, and when x = 6 every cell matches, so A = B.

When the orders differ, the matrices are never equal, whatever the entries. C = [[1, 2, 6], [3, 4, 7]] is 2 × 3 and A is 2 × 2. Their orders differ, so A ≠ C.

To find unknowns, equate the corresponding cells and write the equations. If [[2, 5, 7], [6, 8, −9], [4, 0, 0]] + [[−6, 8, −x], [2, y, −9], [1, z, 2]] = [[−4, 13, 7], [8, 8, −18], [5, 0, 2]], we get 7 − x = 7, 8 + y = 8 and 0 + z = 0. So x = y = z = 0. Now see how the addition itself works.

2. Adding and subtracting cell by cell

For A and B of the same order, each entry of A + B is the sum of the corresponding entries of A and B, and each entry of A − B is their difference. The result has the same order as A and B.

Before adding, lay one matrix over the other. Change the rows and columns of A and B, and watch which cells find a partner and which are left alone.

If the orders are not the same, you cannot add or subtract. Among A (3 × 3), B (3 × 2), C (3 × 1) and D (3 × 3), A − D is possible, but A + B and B − C are not. Sort a few yourself below.

3. Addition can swap, subtraction cannot

A + B = B + A always holds. But A − B = B − A only when A = B; if A ≠ B then A − B ≠ B − A. Take A = [[1, 2], [3, 4]] and B = [[2, −1], [0, 1]]: A + B = B + A = [[3, 1], [3, 5]]. Yet A − B = [[−1, 3], [3, 3]] and B − A = [[1, −3], [−3, −3]].

Example 1 in the book shows the same thing on bigger matrices. For A = [[8, 4, −1], [0, 1, 3], [5, 4, 8]] and B = [[−4, 6, 2], [1, 3, 7], [5, 4, 1]], A + B = [[4, 10, 1], [1, 4, 10], [10, 8, 9]]. A − B = [[12, −2, −3], [−1, −2, −4], [0, 0, 7]] and B − A = [[−12, 2, 3], [1, 2, 4], [0, 0, −7]]. So A + B = B + A, but A − B ≠ B − A.

4. Scalar multiples

For a matrix A and a constant k, kA is the matrix each of whose entries is k times the corresponding entry of A. If A = [[2, 3, 4], [1, 2, 10]] then kA = [[2k, 3k, 4k], [k, 2k, 10k]].

Now work it out yourself. In each cell multiply by k first, then add or subtract. Pick one of the four calculations and write the result cell by cell.

The first two parts of the book's Exercise 1(A) are among these four. For A = [[2, −5, 1], [3, 0, −4]] and B = [[1, −2, −3], [0, −1, 5]], 3A + 4B = [[10, −23, −9], [9, −4, 8]]. For A = [[3, 1, −1], [2, 3, 4], [−4, 5, 6]] and B = [[1, −4, 6], [2, 0, −7], [3, 5, 0]], 7A − 5B = [[16, 27, −37], [4, 21, 63], [−43, 10, 42]].

5. When the sum and the difference are known

If P + Q and P − Q are known, adding gives 2P = (P + Q) + (P − Q), and subtracting gives 2Q = (P + Q) − (P − Q). Then divide by 2. If P + Q = [[3, −1], [−4, 2]] and P − Q = [[5, 7], [6, −2]] then Q = [[−1, −4], [−5, 2]] and P = [[4, 3], [1, 0]].

With this method every unknown comes from the equation of one cell, just as in section 1. Pick one of three problems and enter the unknowns. A wrong one shows below which cell's equation to look at.

6. Check yourself

Three questions in the exam's style. Start with the first.

In the second, look at the orders first.

In the last, match cell by cell.

The whole note on seven cards.

Source: Higher Mathematics 1st Paper, Classes 11-12 (NCTB), Chapter 1 “Matrices and Determinants”, section 1.2.