Finding a, b, c, x, y, z by Equating Matrices
Question:
Find \( a, b, c, x, y, z \) if
\[ \begin{bmatrix} x+3 & z+4 & 2y-7 \\ 4x+6 & a-1 & 0 \\ b-3 & 3b & z+2c \end{bmatrix} = \begin{bmatrix} 0 & 6 & 3y-2 \\ 2x & -3 & 2c+2 \\ 2b+4 & -21 & 0 \end{bmatrix} \]
Concept Used
Two matrices are equal if their corresponding elements are equal.
Step 1: Equate Corresponding Elements
\[ x+3 = 0 \quad …(1) \] \[ z+4 = 6 \quad …(2) \] \[ 2y-7 = 3y-2 \quad …(3) \]
\[ 4x+6 = 2x \quad …(4) \] \[ a-1 = -3 \quad …(5) \] \[ 0 = 2c+2 \quad …(6) \]
\[ b-3 = 2b+4 \quad …(7) \] \[ 3b = -21 \quad …(8) \] \[ z+2c = 0 \quad …(9) \]
Step 2: Solve Variables
From (1): \[ x = -3 \]
From (2): \[ z = 2 \]
From (3): \[ 2y-7 = 3y-2 \Rightarrow y = -5 \]
From (5): \[ a-1 = -3 \Rightarrow a = -2 \]
From (6): \[ 2c+2 = 0 \Rightarrow c = -1 \]
From (8): \[ 3b = -21 \Rightarrow b = -7 \]
Step 3: Verify Remaining Equations
Check (4): \[ 4(-3)+6 = -12+6 = -6,\quad 2x = -6 ✔ \]
Check (7): \[ -7-3 = -10,\quad 2(-7)+4 = -10 ✔ \]
Check (9): \[ z+2c = 2 + 2(-1) = 0 ✔ \]
Final Answer
\[ x = -3,\quad y = -5,\quad z = 2,\quad a = -2,\quad b = -7,\quad c = -1 \]