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Question
If X and Y are 2 × 2 matrices, then solve the following matrix equations for X and Y.
2X + 3Y = `[(2, 3),(4, 0)]`, 3Y + 2Y = `[(-2, 2),(1, -5)]`
Solution
Given that,
2X + 3Y = `[(2, 3),(4, 0)]` ......(1)
3Y + 2Y = `[(-2, 2),(1, -5)]` ......(2)
Multiplying equation (1) by 3 and equaion (2) by 2, we get,
3[2X + 3Y] = `3[(2, 3),(4, 0)]`
⇒ 6X + 9Y = `[(6, 9),(12, 0)]` ....(3)
2[3X + 2Y] = `2[(-2, 2),(1, -5)]`
⇒ 6X + 4Y = `[(-4, 4),(2, -10)]` .....(4)
On subtracting eq. (4) from eq. (3) we get
5Y = `[(6 + 4, 9 - 4),(12 - 2, 0 + 10)]`
5Y = `[(10, 5),(10, 10)]`
⇒ Y = `[(2, 1),(2, 2)]`
Now, putting the value of Y in equation (1) we get,
`2"X" + 3 [(2, 1),(2, 2)] = [(2, 3),(4, 0)]`
⇒ `2"X" + [(6, 3),(6, 60)] = [(2, 3),(4, 0)]`
⇒ 2X = `[(2, 3),(4, 0)] - [(6, 3),(6, 6)]`
⇒ 2X = `[(2 - 6, 3 - 3),(4 - 6, 0 - 6)]`
⇒ 2X = `[(-4,0),(-2, -6)]`
⇒ = `1/2 [(-4, 0),(-2, -6)]`
⇒ X = `[(-2, 0),(-1, -3)]`
Hence, X = `[(-2, 0),(-1, -3)]` and Y = `[(2, 1),(2, 2)]`
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