Grade 11 Matrices: Short Notes

11 ශ්‍රේණිය · 19 පරිච්ඡේදය · න්‍යාස

26%

of Grade 11 term tests include it (16 of 61)

marks in past papers (19 questions)

11 min

full lesson with quick checks

🎯 What you'll learn

State the order of a matrix and name row, column, square and identity matrices
Add and subtract matrices of the same order
Multiply a matrix by a number
Multiply two matrices, row by column, and state the order of the product
Find unknowns by equating corresponding elements

Which kade is cheaper?

Amma needs 3 kg of rice and 2 kg of dhal. Shop X sells them at Rs 220 and Rs 300 a kg; shop Y at Rs 210 and Rs 320. One matrix product, shown below, gives both bills at once: Rs 1260 at X and Rs 1270 at Y. Let's learn how.

Two bills in one line

Quantities × prices = totals for shop X and shop Y.

What is a matrix?

A matrix is a rectangular arrangement of numbers in rows (across) and columns (down), inside brackets.The order is rows × columns. Matrix A below has 2 rows and 3 columns, so its order is 2 × 3 and it has 6 elements.The red element −2 is in row 1, column 3.

Special matrices

Left to right below:• Row matrix: one row (order 1 × 3)• Column matrix: one column (2 × 1)• Square matrix: rows = columns (2 × 2)• Identity matrix I: square, 1s on the leading diagonal and 0s elsewhere. For a 2 × 2 matrix A, AI = IA = A.

Adding and subtracting

You can add or subtract matrices only when they have the same order. Add (or subtract) the corresponding elements, the ones in the same position.Two matrices are equal when they have the same order and every pair of corresponding elements is equal.

✏️ A number times a matrix

Multiplying by a number multiplies every element. For A and B below, find 2A − B.

12A: double every element of A: row 1 is 6, 2; row 2 is 0, 4.
2Row 1 of 2A − B: 6 − 1 = 5 and 2 − 4 = −2.
3Row 2: 0 − (−2) = 2 and 4 − 5 = −1.

✅ 2A − B: row 1 = (5, −2), row 2 = (2, −1)

⚠️ Signs and halves

O/L reports note errors when subtracting matrices or multiplying by a fraction like ½. Two habits help:• Write the scalar product as a full matrix first, then subtract.• Subtracting a negative adds: 0 − (−2) = +2.½ of an element like 5 is 5/2, not 2.

Multiplying matrices

AB exists only if columns of A = rows of B.(m × n) × (n × p) gives an m × p matrix.Each element of AB = a row of A times a column of B: multiply in pairs and add.Row (2 3) × column (4, 5) = 2 × 4 + 3 × 5 = 23.

✏️ Row × column, column × row

Find both products shown. Only about 1 in 5 candidates got the second type right in one O/L paper.

1(1 × 2)(2 × 1) gives 1 × 1: 2 × 4 + 3 × 5 = 23.
2(2 × 1)(1 × 2) gives a 2 × 2 matrix, not a single number.
3Each row (one number) times each column (one number): 2 × 4 = 8, 2 × 5 = 10, 3 × 4 = 12, 3 × 5 = 15.

✅ (23) and the 2 × 2 matrix with rows (8, 10), (12, 15)

⚠️ Not element by element

A recurring O/L error: multiplying matching positions, e.g. 2 × 1, 1 × 0, 3 × 2, 4 × 5. That gives rows (2, 0), (6, 20), which is wrong.Multiplication is always row × column. Run a finger along the row of the first matrix and down the column of the second.

📌 Remember

• Order = rows × columns.• Add or subtract only matrices of the same order, element by corresponding element.• kA: multiply every element by k.• AB needs columns of A = rows of B; (m × n)(n × p) is m × p; each element is row × column.• Equal matrices: equate corresponding elements to find unknowns.

🧾 Quick revision sheet

Order of a matrix	rows × columns
Square matrix	Number of rows = number of columns.
2 × 2 identity matrix I	1s on the leading diagonal, 0s elsewhere; AI = IA = A.
When can A + B be found?	Only when A and B have the same order.
When does AB exist?	When the columns of A = the rows of B.
Order of (2 × 3)(3 × 4)	2 × 4
Is AB always equal to BA?	No. The order of multiplication matters.
(2 × 1) column × (1 × 2) row gives…	a 2 × 2 matrix