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Chapter 4: Problem 9

Prove that $\operatorname{det}(A B)=\operatorname{det}(A) \cdot \operatorname{det}(B)\( for any \)A, B \in \mathrm{M}_{2 \times 2}(F)$.

Short Answer

Expert verified
We proved that \(\operatorname{det}(AB) = \operatorname{det}(A) \cdot \operatorname{det}(B)\) for any \(A, B \in \mathrm{M}_{2 \times 2}(F)\) by first computing determinants of A and B separately, followed by finding their product and then the determinant of AB. Finally, we showed that both expressions for \(\operatorname{det}(AB)\) and \(\operatorname{det}(A)\operatorname{det}(B)\) are the same, confirming our claim.

Step by step solution

01

Compute the determinant of matrix A

We can compute the determinant of a 2x2 matrix A using the formula \(\operatorname{det}(A) = a_{11}a_{22} - a_{12}a_{21}\), where \(a_{ij}\) are the elements of matrix A. Suppose A is given by: $$A = \begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix}$$ Then, $$\operatorname{det}(A) = a_{11}a_{22} - a_{12}a_{21}$$
02

Compute the determinant of matrix B

Similarly, compute the determinant of a 2x2 matrix B using the same formula. Suppose B is given by: $$B = \begin{pmatrix} b_{11} & b_{12} \\ b_{21} & b_{22} \end{pmatrix}$$ Then, $$\operatorname{det}(B) = b_{11}b_{22} - b_{12}b_{21}$$
03

Find the product of the determinants of A and B

Now, we will compute the product of the determinants of A and B: $$\operatorname{det}(A)\operatorname{det}(B) = (a_{11}a_{22} - a_{12}a_{21})(b_{11}b_{22} - b_{12}b_{21})$$
04

Compute the product of matrices A and B

Find the product matrix AB by performing the matrix multiplication: $$AB = \begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix} \begin{pmatrix} b_{11} & b_{12} \\ b_{21} & b_{22} \end{pmatrix} = \begin{pmatrix} a_{11}b_{11} + a_{12}b_{21} & a_{11}b_{12} + a_{12}b_{22} \\ a_{21}b_{11} + a_{22}b_{21} & a_{21}b_{12} + a_{22}b_{22} \end{pmatrix}$$
05

Compute the determinant of the product matrix AB

Using the formula for the determinant of a 2x2 matrix, compute the determinant of matrix AB: $$\operatorname{det}(AB) = (a_{11}b_{11} + a_{12}b_{21})(a_{21}b_{12} + a_{22}b_{22}) - (a_{11}b_{12} + a_{12}b_{22})(a_{21}b_{11} + a_{22}b_{21})$$
06

Simplify the expressions and compare

Simplify the expression of \(\operatorname{det}(AB)\): $$\operatorname{det}(AB) = a_{11}a_{22}b_{11}b_{22} - a_{11}a_{22}b_{12}b_{21} - a_{12}a_{21}b_{11}b_{22} + a_{12}a_{21}b_{12}b_{21}$$ Now, compare this expression to the earlier computed product of determinants \(\operatorname{det}(A)\operatorname{det}(B) = (a_{11}a_{22} - a_{12}a_{21})(b_{11}b_{22} - b_{12}b_{21})\) which simplifies to: $$\operatorname{det}(A)\operatorname{det}(B) = a_{11}a_{22}b_{11}b_{22} - a_{11}a_{22}b_{12}b_{21} - a_{12}a_{21}b_{11}b_{22} + a_{12}a_{21}b_{12}b_{21}$$ Since both expressions are the same, we have successfully shown that: $$\operatorname{det}(AB) = \operatorname{det}(A)\operatorname{det}(B)$$

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Most popular questions from this chapter

Let \(r=\operatorname{rank}(A+B) .\) Find \(2 \times 2\) matrices \(A\) and \(B\) such that (a) \(r < \operatorname{rank}(A), \operatorname{rank}(\mathrm{B})\); (b) \(r=\operatorname{rank}(A)=\operatorname{rank}(B)\); (c) \(r > \operatorname{rank}(A), \operatorname{rank}(\mathrm{B})\).

For \(M \in M_{n \times n}(C)\), let \(\bar{M}\) be the matrix such that \((\bar{M})_{i j}=\overline{M_{i j}}\) for all \(i, j\), where $\overline{M_{i j}}\( is the complex conjugate of \)M_{i j}$. (a) Prove that \(\operatorname{det}(\bar{M})=\overline{\operatorname{det}(M)}\). (b) A matrix \(Q \in \mathrm{M}_{n \times n}(C)\) is called unitary if $Q Q^{*}=I\(, where \)Q^{*}=\overline{Q^{t}}\(. Prove that if \)Q$ is a unitary matrix, then \(|\operatorname{det}(Q)|=1\).

Let \(C\) be the classical adjoint of \(A \in M_{n \times n}(F)\). Prove the following statements. (a) \(\operatorname{det}(C)=[\operatorname{det}(A)]^{n-1}\). (b) \(C^{t}\) is the classical adjoint of \(A^{t}\). (c) If \(A\) is an invertible upper triangular matrix, then \(C\) and \(A^{-1}\) are both upper triangular matrices.

Let \(V\) be the vector space of \(n\) -square matrices over a field \(K\). Show that \(W\) is a subspace of \(V\) if \(W\) consists of all matrices \(A=\left[a_{i j}\right]\) that are (a) symmetric \(\left(A^{T}=A \text { or } a_{i j}=a_{j i}\right)\), (b) (upper) triangular, (c) diagonal, (d) scalar.

Relative to the basis \(S=\left\\{u_{1}, u_{2}\right\\}=\\{(1,1),(2,3)\\}\) of \(\mathbf{R}^{2},\) find the coordinate vector of \(v,\) where (a) \(v=(4,-3)\), (b) \(v=(a, b)\).
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