Real Symmetric matrix: TIFR 2018 Part A, Problem 6

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Understand the problem

[/et_pb_text][et_pb_text _builder_version="3.26.6" text_font="Raleway||||||||" background_color="#f4f4f4" box_shadow_style="preset2" custom_margin="10px||10px" custom_padding="10px|20px|10px|20px"]True or false: Let $A$ be a $3 \times 3$ real symmetric matix s.t $A^6=I$ . Then $A^2=I$ [/et_pb_text][/et_pb_column][/et_pb_row][et_pb_row _builder_version="3.25"][et_pb_column type="4_4" _builder_version="3.25" custom_padding="|||" custom_padding__hover="|||"][et_pb_accordion open_toggle_text_color="#0c71c3" _builder_version="3.26.8" toggle_font="||||||||" body_font="Raleway||||||||" text_orientation="center" custom_margin="10px||10px"][et_pb_accordion_item title="Source of the problem" open="on" _builder_version="3.26.8"]TIFR 2018 Part A, Problem 6[/et_pb_accordion_item][et_pb_accordion_item title="Topic" _builder_version="3.26.8" open="off"]Linear Algebra[/et_pb_accordion_item][et_pb_accordion_item title="Difficulty Level" _builder_version="3.26.8" open="off"]Medium[/et_pb_accordion_item][et_pb_accordion_item title="Suggested Book" _builder_version="3.26.8" open="off"]Linear Algebra, Hoffman and Kunze[/et_pb_accordion_item][/et_pb_accordion][et_pb_text _builder_version="3.23.3" text_font="Raleway|300|||||||" text_text_color="#ffffff" header_font="Raleway|300|||||||" header_text_color="#e2e2e2" background_color="#0c71c3" border_radii="on|5px|5px|5px|5px" box_shadow_style="preset3" custom_margin="48px||48px" custom_padding="20px|20px|20px|20px"]

Start with hints

[/et_pb_text][et_pb_tabs active_tab_background_color="#0c71c3" inactive_tab_background_color="#000000" _builder_version="3.26.8" tab_text_color="#ffffff" tab_font="||||||||" background_color="#ffffff"][et_pb_tab title="Hint 0" _builder_version="3.22.4"]Do you really need a hint? Try it first!

[/et_pb_tab][et_pb_tab title="Hint 1" _builder_version="3.26.8"]First investigate the properties of Real Symmetric Matrices.

The most important property of a real symmetric matrix A is encoded in “Spectral Decomposition” of A.

[/et_pb_tab][et_pb_tab title="Hint 2" _builder_version="3.26.8"]

If matrix $A$ then there exists $Q$ with $Q'Q=I$ such that $A = Q'BQ$ ,where $B$ is a diagonal matrix with diagonal entries being the eigenvalues of A which are real numbers.

Here assume that the eigen values are $a,b,c$ .

We will use this spectral decomposition of A.

[/et_pb_tab][et_pb_tab title="Hint 3" _builder_version="3.26.8"]

Observe that $A^6 = I \Rightarrow Q'(B^6)Q = I$

.(Check!)

$Q'(B^6)Q=I \Rightarrow Q[Q'(B^6)Q]Q'= QQ'= I \Rightarrow B^6=I$

. [as Q is orthogonal]

$B^6$ is a diagonal matrix with diagonal entries real numbers raised to the power 6 i.e $a^6,b^6$ and $c^6$ .

[/et_pb_tab][et_pb_tab title="Hint 4" _builder_version="3.26.8"]

Now hint 3 implies $a^6=1,b^6=1$ and $c^6=1$ .

$a^2=1,b^2=1$ and $c^2=1$ as $a,b$ and $c$ are real numbers. This means $B^2=I$ .

$A^2 = Q'(B^2)Q = Q'Q = I$

.

Hence the given statement is TRUE.

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Watch the video

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Connected Program at Cheenta

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The higher mathematics program caters to advanced college and university students. It is useful for I.S.I. M.Math Entrance, GRE Math Subject Test, TIFR Ph.D. Entrance, I.I.T. JAM. The program is problem driven. We work with candidates who have a deep love for mathematics. This program is also useful for adults continuing who wish to rediscover the world of mathematics.[/et_pb_blurb][et_pb_button button_url="https://www.cheenta.com/collegeprogram/" button_text="Learn More" button_alignment="center" _builder_version="3.23.3" custom_button="on" button_bg_color="#0c71c3" button_border_color="#0c71c3" button_border_radius="0px" button_font="Raleway||||||||" button_icon="%%3%%" button_text_shadow_style="preset1" box_shadow_style="preset1" box_shadow_color="#0c71c3" background_layout="dark"][/et_pb_button][et_pb_text _builder_version="3.22.4" text_font="Raleway|300|||||||" text_text_color="#ffffff" header_font="Raleway|300|||||||" header_text_color="#e2e2e2" background_color="#0c71c3" border_radii="on|5px|5px|5px|5px" box_shadow_style="preset3" custom_margin="50px||50px" custom_padding="20px|20px|20px|20px"]

Understand the problem

[/et_pb_text][et_pb_text _builder_version="3.26.6" text_font="Raleway||||||||" background_color="#f4f4f4" box_shadow_style="preset2" custom_margin="10px||10px" custom_padding="10px|20px|10px|20px"]True or false: Let $A$ be a $3 \times 3$ real symmetric matix s.t $A^6=I$ . Then $A^2=I$ [/et_pb_text][/et_pb_column][/et_pb_row][et_pb_row _builder_version="3.25"][et_pb_column type="4_4" _builder_version="3.25" custom_padding="|||" custom_padding__hover="|||"][et_pb_accordion open_toggle_text_color="#0c71c3" _builder_version="3.26.8" toggle_font="||||||||" body_font="Raleway||||||||" text_orientation="center" custom_margin="10px||10px"][et_pb_accordion_item title="Source of the problem" open="on" _builder_version="3.26.8"]TIFR 2018 Part A, Problem 6[/et_pb_accordion_item][et_pb_accordion_item title="Topic" _builder_version="3.26.8" open="off"]Linear Algebra[/et_pb_accordion_item][et_pb_accordion_item title="Difficulty Level" _builder_version="3.26.8" open="off"]Medium[/et_pb_accordion_item][et_pb_accordion_item title="Suggested Book" _builder_version="3.26.8" open="off"]Linear Algebra, Hoffman and Kunze[/et_pb_accordion_item][/et_pb_accordion][et_pb_text _builder_version="3.23.3" text_font="Raleway|300|||||||" text_text_color="#ffffff" header_font="Raleway|300|||||||" header_text_color="#e2e2e2" background_color="#0c71c3" border_radii="on|5px|5px|5px|5px" box_shadow_style="preset3" custom_margin="48px||48px" custom_padding="20px|20px|20px|20px"]

Start with hints

[/et_pb_text][et_pb_tabs active_tab_background_color="#0c71c3" inactive_tab_background_color="#000000" _builder_version="3.26.8" tab_text_color="#ffffff" tab_font="||||||||" background_color="#ffffff"][et_pb_tab title="Hint 0" _builder_version="3.22.4"]Do you really need a hint? Try it first!

[/et_pb_tab][et_pb_tab title="Hint 1" _builder_version="3.26.8"]First investigate the properties of Real Symmetric Matrices.

The most important property of a real symmetric matrix A is encoded in “Spectral Decomposition” of A.

[/et_pb_tab][et_pb_tab title="Hint 2" _builder_version="3.26.8"]

If matrix $A$ then there exists $Q$ with $Q'Q=I$ such that $A = Q'BQ$ ,where $B$ is a diagonal matrix with diagonal entries being the eigenvalues of A which are real numbers.

Here assume that the eigen values are $a,b,c$ .

We will use this spectral decomposition of A.

[/et_pb_tab][et_pb_tab title="Hint 3" _builder_version="3.26.8"]

Observe that $A^6 = I \Rightarrow Q'(B^6)Q = I$

.(Check!)

$Q'(B^6)Q=I \Rightarrow Q[Q'(B^6)Q]Q'= QQ'= I \Rightarrow B^6=I$

. [as Q is orthogonal]

$B^6$ is a diagonal matrix with diagonal entries real numbers raised to the power 6 i.e $a^6,b^6$ and $c^6$ .

[/et_pb_tab][et_pb_tab title="Hint 4" _builder_version="3.26.8"]

Now hint 3 implies $a^6=1,b^6=1$ and $c^6=1$ .

$a^2=1,b^2=1$ and $c^2=1$ as $a,b$ and $c$ are real numbers. This means $B^2=I$ .

$A^2 = Q'(B^2)Q = Q'Q = I$

.

Hence the given statement is TRUE.

[/et_pb_tab][/et_pb_tabs][et_pb_text _builder_version="3.23.3" text_font="Raleway|300|||||||" text_text_color="#ffffff" header_font="Raleway|300|||||||" header_text_color="#e2e2e2" background_color="#0c71c3" border_radii="on|5px|5px|5px|5px" box_shadow_style="preset3" custom_margin="48px||48px" custom_padding="20px|20px|20px|20px"]

Watch the video

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Connected Program at Cheenta

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The higher mathematics program caters to advanced college and university students. It is useful for I.S.I. M.Math Entrance, GRE Math Subject Test, TIFR Ph.D. Entrance, I.I.T. JAM. The program is problem driven. We work with candidates who have a deep love for mathematics. This program is also useful for adults continuing who wish to rediscover the world of mathematics.[/et_pb_blurb][et_pb_button button_url="https://www.cheenta.com/collegeprogram/" button_text="Learn More" button_alignment="center" _builder_version="3.23.3" custom_button="on" button_bg_color="#0c71c3" button_border_color="#0c71c3" button_border_radius="0px" button_font="Raleway||||||||" button_icon="%%3%%" button_text_shadow_style="preset1" box_shadow_style="preset1" box_shadow_color="#0c71c3" background_layout="dark"][/et_pb_button][et_pb_text _builder_version="3.22.4" text_font="Raleway|300|||||||" text_text_color="#ffffff" header_font="Raleway|300|||||||" header_text_color="#e2e2e2" background_color="#0c71c3" border_radii="on|5px|5px|5px|5px" box_shadow_style="preset3" custom_margin="50px||50px" custom_padding="20px|20px|20px|20px"]