Let ME Mm+n(F) be a block-triangular matrix, that is, an (m+n) x (m+n) matrix composed of...

 

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Let ME Mm+n(F) be a block-triangular matrix, that is, an (m+n) x (m+n) matrix composed of blocks A B - (8|8). B A O (8) (+)) .) In M = Here, two blocks A E Mm (F), CE M₁ (F) are square matrices, and O E Mn,m (F) is the n x m zero matrix (we do not care about the right- uppermost block B E Mmn (F)). Show that det (M) det (A) det (C). (Hint: Prove that M is a product of two block-triangular matrices M = = Let ME Mm+n(F) be a block-triangular matrix, that is, an (m+n) x (m+n) matrix composed of blocks A B - (8|8). B A O (8) (+)) .) In M = Here, two blocks A E Mm (F), CE M₁ (F) are square matrices, and O E Mn,m (F) is the n x m zero matrix (we do not care about the right- uppermost block B E Mmn (F)). Show that det (M) det (A) det (C). (Hint: Prove that M is a product of two block-triangular matrices M = = Let ME Mm+n(F) be a block-triangular matrix, that is, an (m+n) x (m+n) matrix composed of blocks A B - (8|8). B A O (8) (+)) .) In M = Here, two blocks A E Mm (F), CE M₁ (F) are square matrices, and O E Mn,m (F) is the n x m zero matrix (we do not care about the right- uppermost block B E Mmn (F)). Show that det (M) det (A) det (C). (Hint: Prove that M is a product of two block-triangular matrices M = = Let ME Mm+n(F) be a block-triangular matrix, that is, an (m+n) x (m+n) matrix composed of blocks A B - (8|8). B A O (8) (+)) .) In M = Here, two blocks A E Mm (F), CE M₁ (F) are square matrices, and O E Mn,m (F) is the n x m zero matrix (we do not care about the right- uppermost block B E Mmn (F)). Show that det (M) det (A) det (C). (Hint: Prove that M is a product of two block-triangular matrices M = = Let ME Mm+n(F) be a block-triangular matrix, that is, an (m+n) x (m+n) matrix composed of blocks A B - (8|8). B A O (8) (+)) .) In M = Here, two blocks A E Mm (F), CE M₁ (F) are square matrices, and O E Mn,m (F) is the n x m zero matrix (we do not care about the right- uppermost block B E Mmn (F)). Show that det (M) det (A) det (C). (Hint: Prove that M is a product of two block-triangular matrices M = = Let ME Mm+n(F) be a block-triangular matrix, that is, an (m+n) x (m+n) matrix composed of blocks A B - (8|8). B A O (8) (+)) .) In M = Here, two blocks A E Mm (F), CE M₁ (F) are square matrices, and O E Mn,m (F) is the n x m zero matrix (we do not care about the right- uppermost block B E Mmn (F)). Show that det (M) det (A) det (C). (Hint: Prove that M is a product of two block-triangular matrices M = =

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To prove that detM detAdetC we will show that M can be written as the product of two block... View the full answer