Variance calculation in matrix notation for $var(z-Ax)$

Question

I noted from a post here that

$$var(z - \mathbf{A}x)=var(z)+var(\mathbf{A}x)-\mathbf{A}cov(z, -x)-cov(z,-x)\mathbf{A}^T (Eq. 1)$$ (I dropped the conditional part in the original formula from the post since I believe it irrelevant to my question. If I write the above formula incorrectly, please refer to the original formula in the aforementioned post.)

Additionally, I noted: $$var(x_1 + \mathbf{A}x_2) = var(x_1)+\mathbf{A}var(x_2)\mathbf{A}^T+\mathbf{A}cov(x_1, x_2)+cov(x_2,x_1)\mathbf{A}^T (Eq. 2)$$

I do not particularly understand the $\mathbf{A}cov(z, -x)-cov(z,-x)\mathbf{A}^T$ part in $Eq.1$ and the $\mathbf{A}cov(x_1, x_2)+cov(x_2,x_1)\mathbf{A}^T$ part in $Eq.2$.

I wish to see a step-by-step proof of how the two $cov$ components are derived, particularly for $Eq.1$. (No proof of $Eq.2$ is needed if the proofs are similar.)

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I followed User1865345's approach for Eq1:

$$ var(z-Ax) = E[ [(z-\mu_z)-A(x-\mu_x)] [(z-\mu_z)-A(x-\mu_x)]^T ] = E[ (z-\mu_z)(z-\mu_z)^T - (z-\mu_z)(x-\mu_x)^TA^T - A(x-\mu_x)(z-\mu_z)^T + A(x-\mu_x)(x-\mu_x)^TA^T ] = var(z) - cov(z, x)A^T - Acov(x, z) + var(Ax) $$

Unfortunately, the above result is not identical from Eq.1. As I mentioned in my question, I did omitted conditional expression in Eq1 from the original post. I wish to know if there is anything wrong with my proof above.

Answer 1

Discussing about $\mathrm{Eq}. ~(2),$ with $\mathbf X_i, ~\mathbb E\mathbf X_i:=\boldsymbol\mu_i, ~i\in\{1, 2\}$ and $\bf A$ being a non-stochastic matrix,

\begin{align} \mathbb{Var}(\mathbf X_1 + \mathbf{ AX_2}) &= \mathbb E[(\mathbf X_1-\boldsymbol \mu_1) + \mathbf A(\mathbf X_2-\boldsymbol\mu_2) ][(\mathbf X_1-\boldsymbol \mu_1) + \mathbf A(\mathbf X_2-\boldsymbol\mu_2) ]^\mathsf T\\ &= \mathbb E\left[(\mathbf X_1-\boldsymbol \mu_1) (\mathbf X_1-\boldsymbol \mu_1) ^\mathsf T+\mathbf A(\mathbf X_2-\boldsymbol\mu_2)(\mathbf X_1-\boldsymbol \mu_1) ^\mathsf T+(\mathbf X_1-\boldsymbol \mu_1)(\mathbf X_2-\boldsymbol\mu_2)^\mathsf T\mathbf A^\mathsf T + \mathbf A(\mathbf X_2-\boldsymbol\mu_2)(\mathbf X_2-\boldsymbol\mu_2)^\mathsf T\mathbf A^\mathsf T\right]\\&= \mathbb {Var}(\mathbf X_1) +\mathbf A\mathbb{Cov}(\mathbf X_2, \mathbf X_1) +\mathbb{Cov}(\mathbf X_1, \mathbf X_2)\mathbf A^\mathsf T+ \mathbf A\mathbb{Var}(\mathbf X_2) \mathbf A^\mathsf T. \end{align}