Introduction: A straightforward topic if you follow the rules. Matrix multiplication is the most complex part – remember the "row-by-column" rule. Pay attention to the conditions for when operations are possible (e.g., same order for addition).

The "S Pass" Foundation (නියත S එකකට)

• Prompt 1 (Define): What is the 'order' of a matrix? Give the order of A=(10​5−2​34​).

• Prompt 2 (Calculate): Given A=(21​50​) and B=(−12​34​), find A+B.

• Prompt 3 (Calculate): Using the matrices above, find 3A.

• Prompt 4 (Identify): What is a 'unit matrix'? Write down the unit matrix of order 2.

• Prompt 5 (Method): For two matrices P and Q, what condition must be met to find the product PQ?

Climbing to a "C" (C එකට පාර)

• Prompt 1 (Calculate): If P=(4−1​23​) and Q=(10​5−2​), calculate 2P−Q.

• Prompt 2 (Solve): Find the values of a, b, and c if (a−1​32​)+(4c​b0​)=(52​72​).

• Prompt 3 (Multiply): Find the product AB where A=(32​14​) and B=(15​0−2​).

• Prompt 4 (Multiply): Is the product BA for the matrices above the same as AB? Calculate it to check.

• Prompt 5 (Application): A store sells two types of cakes, A and B. On Monday, they sell 10 of A and 15 of B. On Tuesday, they sell 12 of A and 20 of B. Cake A costs Rs. 80 and Cake B costs Rs. 100. Use matrix multiplication to find the total revenue for each day.

Aiming for a "B" (B ඉලක්කය)

• Prompt 1 (Problem Solve): Given M=(21​−30​), find the matrix M2−2M+I, where I is the unit matrix of order 2.

• Prompt 2 (Solve): Find the 2×2 matrix X such that 3X+(24​−15​)=(81​211​).

• Prompt 3 (Explain): If P is a 3×2 matrix and Q is a 2×4 matrix, what is the order of the product PQ? Is the product QP defined? Explain why.

• Prompt 4 (Inverse): The inverse of a 2×2 matrix A=(ac​bd​) is A−1=ad−bc1​(d−c​−ba​). Find the inverse of A=(31​21​).

• Prompt 5 (Using Inverse): The simultaneous equations 3x+2y=7 and x+y=3 can be written in matrix form as (31​21​)(xy​)=(73​). Using the inverse from the previous prompt, solve for x and y.

Securing the "A" Distinction (A සාමාර්ථය තහවුරු කරගන්න)

• Prompt 1 (Challenge): Find a 2×2 matrix A such that A(13​24​)=(57​68​).

• Prompt 2 (Proof): Show that for any two 2×2 matrices A and B, it is not always true that AB=BA. Provide a counterexample.

• Prompt 3 (Determinant): The value ad−bc is the determinant of a 2×2 matrix. A matrix has no inverse if its determinant is 0. For what value of k does the matrix (k2​4k−2​) have no inverse?

• Prompt 4 (Synthesis): A transformation in a coordinate plane is represented by the matrix T=(01​−10​). Find the coordinates of the image of the triangle with vertices (1,1), (3,1) and (3,4) under this transformation. Describe the transformation geometrically.

• Prompt 5 (Problem Solve): If A=(12​−1−1​) and B=(14​xy​), and (A+B)2=A2+B2, find the values of x and y.