Example 10: Find $x$ and $y$ from a matrix equation

Basics you need

Solution

1. Check the orders

   All three matrices have 2 rows and 2 columns, so all are $2\times2$⁠. The scalar multiple keeps the order, the sum of two $2\times2$ matrices is defined and is $2\times2$⁠, and it can be compared with the $2\times2$ right side.

   Both the sum and the equality need matching orders. If the right side were, say, $2\times3$⁠, no values of $x, y$ could make the equation true.

2. Multiply the first matrix by 2

   Every entry is doubled, including those with unknowns:

   $$\begin{aligned} 2\cdot x &= 2x, & 2\cdot5 &= 10,\\ 2\cdot7 &= 14, & 2(y - 3) &= 2y - 6. \end{aligned}$$ $$2\begin{bmatrix} x & 5 \\ 7 & y-3 \end{bmatrix} = \begin{bmatrix} 2x & 10 \\ 14 & 2y-6 \end{bmatrix}.$$
   A scalar multiplies every entry. In the entry $y - 3$⁠, the 2 must multiply both terms of the bracket: $2(y - 3) = 2y - 6$⁠, not $2y - 3$⁠.
   $kA = [k\,a_{ij}]$ (Section 3.4.2); the distributive law for numbers, $2(y - 3) = 2y - 6$⁠. Lesson: addition and scalar multiplication

3. Add the second matrix

   Add corresponding entries:

   $$\begin{aligned} (1,1):&\quad 2x + 3\\ (1,2):&\quad 10 + (-4) = 6\\ (2,1):&\quad 14 + 1 = 15\\ (2,2):&\quad 2y - 6 + 2 = 2y - 4 \end{aligned}$$

   So the left side is

   $$\begin{bmatrix} 2x+3 & 6 \\ 15 & 2y-4 \end{bmatrix}.$$
   Matrix addition works position by position. In the entries with unknowns we can only combine the number parts: $2x + 3$ stays as it is, and $-6 + 2 = -4$⁠.
   $A + B = [a_{ij} + b_{ij}]$ (Section 3.4.1). Basics: adding only like with like

4. Equate corresponding entries

   The equation now reads

   $$\begin{bmatrix} 2x+3 & 6 \\ 15 & 2y-4 \end{bmatrix} = \begin{bmatrix} 7 & 6 \\ 15 & 14 \end{bmatrix}.$$

   By equality of matrices, entry by entry:

   $$\begin{aligned} (1,1):&\quad 2x + 3 = 7\\ (1,2):&\quad 6 = 6 \ \text{(true)}\\ (2,1):&\quad 15 = 15 \ \text{(true)}\\ (2,2):&\quad 2y - 4 = 14 \end{aligned}$$
   This answers the textbook's "(Why?)": two matrices are equal exactly when their corresponding entries are equal, so the matrix equation gives these four number equations. Two of them have no unknown and are true, so they cause no contradiction.
   Definition 2, Section 3.3.1. Lesson: equality of matrices

5. Solve for $x$ and $y$

   $$\begin{aligned} 2x + 3 = 7 &\ \Rightarrow\ 2x = 4 \ \Rightarrow\ x = 2,\\ 2y - 4 = 14 &\ \Rightarrow\ 2y = 18 \ \Rightarrow\ y = 9. \end{aligned}$$
   For $x$⁠: subtract 3 from both sides, then divide by 2. For $y$⁠: add 4 to both sides, then divide by 2. Each equation has exactly one solution.
   Undo the operations in reverse order. Basics: solving equations

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