(b) Given that $$P = \begin{bmatrix} 2 & -1 \\ 3 & 4 \end{bmatrix}, \quad Q = \begin{bmatrix} 5 & 2 \\ 7 & 4 \end{bmatrix} \quad \text{and} \quad R = \begin{bmatrix} 2 & 5 \\ 3 & 8 \end{bmatrix},$$ find a matrix $S$ such that $PQ - RS$ is a null matrix. (matrices, matrix multiplication, null matrix, matrix equation)

Step 1: Set up the matrix equation

The problem states that $PQ - RS$ is a null matrix. A null matrix, denoted by $O$, is a matrix where all its elements are zero. We can rewrite the equation to solve for $RS$.

Step 2: Rearrange the equation

To isolate the term involving $S$, we move $RS$ to the right side of the equation. This makes the equation $PQ = RS$, which we will use to find $S$.

Step 3: Calculate PQ

First, we need to calculate the product of matrices $P$ and $Q$. Matrix multiplication involves multiplying rows of the first matrix by columns of the second matrix. Let's compute each element of the resulting matrix.

Step 4: Perform matrix multiplication for PQ

We perform the element-wise multiplications and additions: $(2)(5) + (-1)(7) = 10 - 7 = 3$, $(2)(2) + (-1)(4) = 4 - 4 = 0$, $(3)(5) + (4)(7) = 15 + 28 = 43$, and $(3)(2) + (4)(4) = 6 + 16 = 22$. Thus, we get the matrix $PQ$.

Step 5: Set up the equation for S

Now we have the equation $RS = PQ$. We know $R$ and $PQ$, and we need to find $S$. Since $R$ is a $2 \times 2$ matrix and $PQ$ is a $2 \times 2$ matrix, $S$ must also be a $2 \times 2$ matrix. Let's represent $S$ with unknown elements.

Step 6: Assume S and form the equation

Let $S = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$. Now we can substitute the matrix $R$ and the unknown matrix $S$ into the equation $RS = PQ$ and perform the matrix multiplication on the left side.

Step 7: Perform matrix multiplication for RS

We multiply the matrix $R$ by the matrix $S$. This results in a new matrix whose elements are expressions of $a, b, c,$ and $d$. This matrix is equal to the matrix $PQ$ we calculated earlier.

Step 8: Form a system of linear equations

By equating the corresponding elements of the matrices on both sides, we obtain four linear equations. We can solve these systems of equations to find the values of $a, b, c,$ and $d$.

Step 9: Solve for a and c

We solve the first system of two equations for $a$ and $c$. We multiply the first equation by 3 and the second by 2 to make the coefficients of $a$ equal. Subtracting the modified first equation from the second yields $c = 77$. Substituting this value back into the first equation allows us to find $a = -191$.

Step 10: Solve for b and d

Next, we solve the second system of two equations for $b$ and $d$ using the same method. Multiplying the first by 3 and the second by 2, then subtracting them, results in $d = 44$. Substituting this back into the first equation gives $b = -110$.

Step 11: Formulate matrix S

With the values of $a, b, c,$ and $d$ found, we can now construct the matrix $S$. This matrix satisfies the original condition that $PQ - RS$ is a null matrix.

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