Matrix Solution

Question

I have matrix integral equation of the following form

${f^{'}(x)}_{1 \times 1}A_{3\times 3}=P_{3\times3} (1-x)+Q_{3 \times 3}x \tag 1$ .

All dimensions are indicated in equation itself. " ' " indicate derivative with respect x

Data given

1. $P,Q$ are constants,and non invertible
2. A is a constant
3. $f(x)$ is a scalar function

Question

What is the expression for $f(x)A \tag 2$. Simple integration appears to me. But I am just confused with matrix notation. I am new to that. Asking for confirmation. Thanks

Answer 1

You can integrate both sides just as if you had only numbers, so $$ f(x)A=P(x-\frac12x^2)+\frac12Qx^2+C, $$ where $C$ is a constant matrix.

A more generally applicable remark: If you want to solve the scalar function $f$ from your equation, you can do it without worrying too much about matrices. If $A$ is the zero matrix, then $f$ can be anything. If $A$ is nonzero, there is a nonzero element $a_{ij}$ (perhaps $i=2$ and $j=1$). Your equation is an equation of matrices, so each element must be equal on both sides. Thus you get $$ f'(x)=\frac{p_{ij}(1-x)+q_{ij}(x)}{a_{ij}}. $$ This is a scalar equation and easy to solve. If you want to find $f(x)A$, just multiply the so found solution $f$ by $A$.